smaller tires at the same pressure roll faster due to less resistance, and the way that BMX is ridden it is very noticable
smaller tires are lighter yes, assuming everything else is the same (i.e. tire tread pattern, kevlar vs steel beads etc)
i'd argue, however, that "kids now a days go through hubs and parts faster than we used to" can be due to a lot of factors. we went thru an era where parts weren't well made and just... die after a while, and you mcgyvver which ever way you can do keep the bike running. then came the era of heavier bikes, durability being priority (alex triple wall rims, 9 lkb pork frames). as we coasted thru the 36t sprocket to smaller and lighter, parts became better made overall. but after a certain point, it's a choice of losing a bit more durability to get a lighter bike, with a higher expected frequency of parts replacement. e.g. how much heavier would a 48 spoke wheel be, compared to 36? it's 12 spokes, bigger flanges on the hub, and a few more holes on teh rim (plus some minute weights here and there). but nobody rides 48 now because 36s are deemed strong enough, and everyone prefer a slight lighter wheel with the understanding that you'll replace it a bit more frequently.
now take that philosophy and apply it to the rest of the bike - now you replace parts a bit more frequently, but your bike is lighter and more fashionable since you can keep up with the fresh cool colour ways - companies obviously want to market that way as well... new generation of product every 9 months, with the expectation of people 'upgrading' everyone 18-24 months. but i really digress here.
it really depends on the average coasting speed/ style of your riding really, i think as we grow older towards mid 30's, our joints are working a bit worse and you'd want a more foregiving bike - a fatter tire can mean running slightly lower pressure, and if you're not always riding at full pedal speed, it's something worth considering
note: since i ride a mixture of flatland/ a bit of street whenever i have to clear the spider webs on the bike, i keep 1.75 rear tire with high pressure, and a 1.9 front with high pressure. it's a bit too janky on the front end riding street, and a bit too wide riding flatland, so i'm on a perfect unhappy medium