But without me forcing the game to have 6-8 encounters per long rest, he feels comfortable only having the 3 rages per day he has now. The rage damage also doesn't go up by much from 5th to 11th level (only +1), and we're not planning on hitting level 20 so the capstone doesn't matter.
First, I was just taking issue with the claim that barbarian is a defensive rather than offensive class. I'm not necessarily saying your player shouldn't multiclass.
That said, the rage damage isn't a big deal by itself, but he's adding it up to three times per round with PAM or GWM bonus attack, four times with a reaction attack, and five if he can get a kindly wizard to cast
haste on him (a wizard with
haste probably should in any fight that really matters). Even without 6+ encounters per day, it's nice to be able to rage again when you get knocked out of it by mobile enemies, spells, unconsciousness, etc. This is dependent on your DM and campaign style to a large extent, but it's really not rare in my experience to situationally lose rage.
Rage+Reckless Attack+GWM exists as of level 2; level just lets you do it more. Am I missing something in their higher level scaling? Brutal Critical doesn't seem like it's going to add up to too much.
Feral Instinct at 7, Totem Attunement at 14, and Persistent Rage at 15 are all great. Also, d12 HD every level is great for the barbarian's play style, but here the opportunity cost is obviously dependent on what he would multiclass into. I haven't calculated the damage increase from Brutal Critical. My guess is that by itself it would be outweighed by available damage increasers from several MCs. In particular, the barbarian has really no at-will nova ability.
Finally, outside combat, the barbarian actually has some utility, especially in a campaign with any emphasis on exploration and wilderness adventuring. As long as he has those rages (for resistance), it's no trouble to stick with breastplate and therefore remain Stealth-capable. With his best-in-class hit points, Danger Sense, Spirit Seeker, Fast Movement, Eagle Aspect, Feral Instinct, and Spirit Walker at 10, he's a really excellent scout. He can still be a great scout if he multiclasses out earlier, especially if he chooses rogue or ranger, but I'd be pretty tempted to stick around for Feral Instinct, at least (of course, then he'll get another ASI if he stays until 8, and so on...).
But really, unless you're requiring him to plan it out ahead of time, his best bet is probably just to start playing and see how it goes. Maybe he'll find out he likes the play style and really would like some more rages after all. Or maybe he'll confirm that he has what he wants out of the class by 5 and look to multiclass. I won't tell him he's wrong.
