Required? Not per se.

Due to its features, barbarian gravitates towards the weakest type of character - namely, STR melee martial - while still needing other stats, having extremely limited options for anything other than melee STR combat, the worst scaling in the game, largely horrible subclasses and ultimately middling damage and defenses.

What does that translate into? Well, for the average barbarian, ASIs are always quite the investment. Because the class goes downhill after tier 1 and there's plenty of bases to cover, even for the things often assumed to be the barbarian's schtick, doing something like boosting Str or Dex or Con, or going for some other option meant to focus more on survivability, is plenty viable, simply because, with how limited the barbarian is, any help is welcome.

Being so ASI-starved also means that traditional feat combinations, whose caveats and tradeoffs can be largely ignored on something like a fighter, actually leave more pronounced weaknesses behind. Go too hard on the damage part and you'll end up being unexpectedly squishy, owing to limited rages and the class' otherwise limited defenses combined with a dangerous playstyle. On top of that, there's the added conflict of rage and PAM, which, while not the worst thing ever, is definitely something you could do without. GWM does synergize very well with Reckless alright, but see above about defenses; go too hard on Reckless and you will get your face beaten in if the game's on the less forgiving side.

Then there's a rather funny coincidence, which was highlighted by other posters above. The best barbarian subclasses are ironically far from requiring the aforementioned feats, at least in such bulk. The Zealot's damage rider means diminishing effects for GWM and when it comes to PAM you have an extra bonus action to consider later on, though at least that part is rather insignificant due to the 1/long rest restriction said bonus action has. Beast works with its own weapons and therefore doesn't synergize with weapon feats. And Ancestral Guardian is more defensively focused on top of having a dedicated reaction, so GWM is not as important to them while PAM and/or Sentinel require some consideration before picking due to contesting the aforementioned reaction (albeit here one of the two could arguably work as a complimentary effect to Spirit Shield).

So, required? Nah. If you do want to go balls to the wall with damage and are utilizing a middle-of-the-pack or worse subclass then yeah, it would be a good idea to select these feats. But there's subclasses and playstyles that don't rely on these and arguably those subclasses and playstyles are superior for reasons beyond giving you more flexibility with your ASIs, so you're far from condemned to PAM+GWM/Sentinel.