There's little doubt that, for most people, wearing a belt boosts the amount they can squat. As such, if you squat just to build legs then, I guess, there's little to suggest you shouldn't wear a belt since greater weight gives greater stimulus. You could apply the same argument to using a vertical leg press, too.
When it comes to being strong, though, and having functional strength then you need to know how to form a tight core without a belt. You use your abdominal muscles in very different ways belted and not-belted. When belted you push against the belt to create intra-abdominal pressure to create stability. When not belted you tighten the abs to squeeze the abdominal cavity to create pressure.
Regarding straps, if you really don't care about being able to hold onto things then there's no need to have a strong grip. I use straps when doing heavier ballistic shrugs and heavier rack pulls and that's about it.
I've seen many guys at the gym wrapping straps around the bar as a matter of course for every pulling movement, including warmup rows. Clearly, you don't want a weak grip to compromise a deads workout but a deadlift session is also a chance to improve grip strength. Don't use the straps until your grip is failing and use double overhand whenever that suffices. Your grip will improve.
If I need to move a log out of the way or carry it back to the house for firewood, I like the fact that I don't need to run home for my belt and that I can grip it and expect to hold on.