I've had a fair amount of situations where I've been unable to protect my ADC or any other squishy from an assassin because of clever play on their part. Perhaps we overextend, they focus me down in a couple seconds due to having no budget, and then my ADC loses all of their "tankiness" with me gone and dies since they no longer have anyone to keep the assassin off 'em.
I've had times where I've been the assassin to jump in, clear the squishy with or without their support, then jump out.
I've also had plenty of times where I HAVE saved my carry, and many others where I was unable to get the kill despite my damage as the assassin.
Supports don't just tack on a huge amount of extra health at the push of one button, and even if we do have a lot of shielding the individual ones are weak due to low options for scaling. You can tear through a Rakan E in a couple of autos, tops. Face of the Mountain is only really useful for the slow. Locket is good, but still really easy to fight through on a supp with only sightstone.
I don't think it's that big of an issue in reality, even if it can be vastly exaggerated on paper.