Investing involves risk. Sometimes that means losing money, even if you are the bank.
The reason a $100 par bond paying 2% sells for (I don't know, say) $87 when interest rates are (I don't know, say) 5% isn't that the original bond is impaired. It's that the same $100 buys you a bond that pays 3% better, so nobody will buy the bond without a discount.
But the bank doesn't normally sell the bond to begin with. That's why people say banks "borrow short and lend long". What the bank is supposed to do is hold the bond until it matures and is paid back in full. The only reason SVB can't do that is that all its depositors simultaneously demanded their money bank, so it couldn't wait the bonds out. But other institutions can do that waiting.
It will be a loss for the bank, but that seems better than total collapse. Banks are ultimately companies that take calculated risk to make money, if you can't afford to take an occasional loss then you shouldn't be in a risk based business.
The thread here asks: "who's paying to cover SVB's uninsured depositors?". Isn't that the answer?
Maybe enough depositors will leave money in the SVB at below-market interest rates that it will earn its way out of the hole. The FDIC has given depositors a special incentive to, since by guaranteeing all funds they've made the SVB the safest bank in the USA. If the depositors don't, then the FDIC will take the loss, and socialize it over all participating banks.
Per my other comment, the HTM accounting is a distraction. That accounting was compliant, but accounting doesn't define reality. The holders of long-term bonds take a real economic loss when interest rates increase, regardless of whether they sell. This may seem unintuitive since the cash flows don't change, but it couldn't be otherwise--if the bond is worth par, then why aren't any buyers willing to pay that?
As long as the holder does not intend to sell the bond, believes that is more likely than not going to be a position where it isn't forced to sell (to generate working capital etc), and there is no likelihood of a credit loss, then the bond is not OTTI.
The subjective assessment of whether you're "more likely than not" going to be forced to sell the bond is the pivot on which this whole thing tilts. It's probably a good question whether a simple balance of probabilities is really where that standard ought to be.
If you're willing to lock the depositors up for 10 years and pay them back when those assets mature, sure. But that probably wouldn't be seen as an acceptable way of making those depositors whole.
Someone's got $100 of deposits with you today; you're holding a 10-year bond that will pay $102 over the 10 years but currently trades at $87. Yes you "can" "pay" "them" "back" eventually, but what if they want to pull their deposit today, perhaps to buy a bond like the one you were holding? If you do the accounting based on today, they're entitled to $100 and you only have $87; if you do the accounting based on 10 years' time, they're entitled to $115 and you only have $102; either way there's a shortfall.