I’m calling this one a three-star installment for having a reasonably solid character/emotional core, surrounded by a whole lot of nonsensical plot contrivances. It’s also, of course, the show’s farewell to Wesley Crusher (but for his three guest appearances in future seasons). Wes was always a mixed bag of a character, to me; I never hated him, but I never loved him, either. Replicating that situation in microcosm to an extent, I like him fine in this episode, but I also typically, when re-watching TNG, feel ready to be done with him by the time season four rolls around. It would be cynical in a way that doesn’t sit right with me to award points to “Final Mission” simply for being the episode that ushers Wesley out the door, but I will say that the character story told here works better in the context of his imminent departure for Starfleet Academy than it would have outside of that context, and that’s something.
First, though, the episode has to overdose on nonsense and contrivance in order to get Picard and Wesley into the isolation and crisis scenario that gives rise to their character moments. This begins with the weirdness of transporting them to the mining negotiations via the miners’ ancient and dubious shuttle. Why wouldn’t they just take an Enterprise shuttlecraft? (Or, honestly, the transporter; the Enterprise is apparently in the destination planet’s solar system at the outset of the episode—hence the convenient availability of a moon to crash-land on—so it surely wouldn’t have taken much time to zip into orbit and beam them down before heading off at warp 6 to deal with the radiation barge.) Next, about five minutes after the said dubious craft has somehow passed Geordi’s “safety and operational inspections,” it nevertheless suffers a serious malfunction that forces an emergency crash-landing. The Enterprise has conveniently been called away to deal with an emergency and thus is unable to immediately rescue them—which by itself would be fine, except for how amazingly stupid the entire radiation barge subplot turns out to be (which I’ll get to). Picard, Wesley, and Dirgo crash on an inhospitable desert moon, yet are conveniently within a day’s walk of some mountains that promise shelter from the sun. Upon reaching them, they seemingly immediately locate some very implausible-looking caves, and proceed to converse about whether caves mean water, or if they might have been formed by volcanic activity—all while ignoring the handy stairs leading down into the “planet hell” set from the entrance, which clearly indicate that the caves were in fact created by set designers. Or, okay, maybe by intelligent locals, but even after presenting the characters with a fountain that is inexplicably protected by a force field as a puzzle to solve, the episode never bothers to follow up on how or why any of this is here, or whether it makes any sense. And even at this point, two acts of outrageous stupidity on the part of shuttle captain Dirgo are still required in order to critically injure Picard and then take himself out of the picture, so as to isolate Picard and Wesley with the captain incapacitated and in danger of dying.
Meanwhile, the Enterprise is busy towing a garbage scow full of radioactive waste away from a helpless planet. On top of the issue that this plot would not exactly be riveting even if handled competently, it quickly becomes a laughingstock. An asteroid belt lies between them and the sun, so the barge allegedly has to be navigated between the asteroids rather than simply shoved in the desired direction. They couldn’t direct it around the asteroid belt, via a series of quick pushes? Or just direct it away from the planet and off into the void of deep space, rather than specifically toward the sun? (It could then have been redirected or destroyed later, if important.) Or, I don’t know…maybe even clear a path through the asteroid belt with phasers, or something? There’s just nothing plausible about the predicament that we’re meant to see the stupid radiation barge as posing. Add to that another one of TNG’s absurd countdowns to lethal radiation exposure (with zero harmful affects being experienced prior to the end of the countdown), and you get a B plot that just has nothing going for it at all.
But, okay. Wesley has been accepted to the academy, and Captain Picard, his mentor and surrogate father figure, asks him to accompany him on a “final mission” before his departure. The episode opens with Picard having some fun at Wesley’s expense by play-acting the stern disciplinarian, then puts the two of them in a situation that forces Wesley to “step up” and save them both (but in particular, the injured captain) on his own. By the end, when Picard feebly gestures toward a repeat of the opening joke, Wesley lobs it back at him this time; he’s achieved, at least for the moment, a kind of equality with his mentor. These are some serviceable story beats. Wesley, while trying to keep Picard awake, calling back to the previous occasion when he and the captain went on a shuttle ride together is also nice, and his admission that making Picard proud has always been central to what drives him reaches to the core of what makes the Picard-Wesley character pairing work. Meanwhile, Picard’s line about how he envies Wesley for being poised “just at the beginning of the adventure” is wonderful. Picard is facing the fact that he may be dying, and “the adventure” to which he refers, of course, is life. There’s definitely something a bit…boyish?…about our distinguished captain still envisioning life as an “adventure,” but it totally fits with the insights that we gained about Jean-Luc Picard in “Family,” and I find it utterly believable that this bit of character truth spills out on what might have been his deathbed—and even more particularly, that he voices it to Wesley. (As usual, character insights more far-reaching than my own can be found here, courtesy of William B; see the first comment under the review.) So I like these things about “Final Mission.” I also appreciate that Picard regrets the “selfishness” that impelled him to bring Wesley along on the trip. Finally, the fact that the outcome of Wes’s determination to emulate his mentor and make him proud one more time amounts to him Starfleeting the hell out of their predicament by cooking up an improvised-tech solution to the force field problem feels fitting. It’s sort of adorably in character for the series, but it also feels here like it actually serves story rather than just being a plot device.
I will say, though, that I think there was potential for a slightly more focused Wesley character story, which I would have appreciated even more—but it foundered mainly on the rocks of some clumsy writing where the uninspired character of Dirgo was concerned. Passing over the fact that Dirgo is, in the words of the reviewer whose site I linked to above, “too stupid to live,” my main complaint is about the incoherent dynamic between him and Wesley. There’s a nice moment when the incapacitated Picard impresses upon Wes that it’s up to him to “handle” Dirgo; “you’re going to have to stand up to Dirgo on your own,” he says. The thing is, up to this point Wesley has already been calling Dirgo on his dipshitery at every turn. In fact, he’s been doing so too stridently, really; if there was a lesson that he needed to learn here from his captain, it was about how to use diplomacy to defuse a loose cannon like Dirgo (as the captain himself does when Dirgo is pitching a fit over Picard deciding that they need to hike to the mountains). In this context, the line about standing up to Dirgo doesn’t quite make sense. If the episode had thus far shown us Wesley being cowed by Dirgo, this would have tracked better—or, for a more nuanced approach, the lesson could even have been that earning Dirgo’s respect in the way that Picard had tried to do (instead of berating and ridiculing him, as Wesley had been doing) would serve Wesley later, when he had to stand up to the idiot and stop him from doing something unbelievably stupid. Instead, what we got is a bit of incoherence that kind of leads nowhere, with Wesley failing to stop Dirgo from getting himself killed, and not even taking any particularly meaningful lesson away from the experience (other than “I should have tried harder”). It definitely seems like an opportunity was missed, here.
In any case, what we have overall with “Final Mission” is some serviceable character material wrapped in a bunch of garbage. For all its nonsensicality, though, the plot stupidity is not offensive enough to negate the Picard/Wesley story at the episode’s center, which preserves this as a nominal “keeper,” in my book.

“There’s just nothing plausible about the predicament that we’re meant to see the stupid radiation barge as posing”
To add to that, actual asteroid belts are not as typically depicted in sci fi. They are almost entirely empty space. It would be a bizarre phenomena indeed that would stop a barge from getting through one, especially to a civilization with the level of spaceflight mastery and computer automation that Trek depicts. Tractor beams, transporters, phasers, and automated crafts would all seem to present ready solutions here, though I’m sure the episode handwaves a bunch of that away.
Completely agree with your review here. I consider it worthwhile mostly for Picard’s interaction with Wes and the somehow-not-cheesy “just at the beginning of the adventure” line, pulled off by the masterful Patrick Stewart.