Well, I'd argue firstly that Health is much more important that armor/MR. Obviously it's your effective health that matters most, and at any given amount of health or armor/MR one of the two will grant more effective health than the other (and thus in many cases you do actually want armor/MR).
All I'm saying is, you're not going to be very tanky if you're building items like Frozen Heart and Thornmail. You'll be a LOT tankier with Randuins Omen and Sunfire cape for example. Or Spirit Visage and Banshee's Veil for MR.
Of course the counter to stacking health is BotRK or Liandry's (percent health damage). At least in that case though, you're forcing the enemy to build a whole item just to counter you (and only that person will do much damage). At these points you're not looking to tank in a teamfight for days though, you're looking to survive long enough to apply your CC in the fight so your carries can take out their carries and not the opposite.
As a closing comment, I want to state that you need to pay attention to gold differences. If you're underfed as a tank you will and should be getting wrecked quickly by a fed carry. I've seen plenty of tanks survive for absurd amounts of time when they're even or ahead. It feels like most of these "carries are OP, tanks suck" rants are due to tanks who don't have very good CS or KDAs and are getting blown up by a carry with 2k+ more gold than them (or tanks that are itemizing incorrectly).