They were stuck with assets earning <2% for 10+ years, meanwhile depositors were demanding higher returns (else they would put their money in other banks, in 6m tbills, 10yr treasuries, etc.. all of which paid a lot more).
And thus, "it's not a solvency issue as long as depositors are stupid and don't realize they can get more $$$ from other banks" ends up being quite misleading
The markets can stay irrational longer than you can remain solvent.
If a firm's assets will eventually mature and be worth more than the current liabilities, then a private rescuer can make a lot of money by bailing them out. If the assets will never recover or pay out, then someone will be holding the bag.
Mark to market is real. If you need liquidity immediately from the market, whether through a rescuer or otherwise, you do your accounting using that value and in SVB's case they were insolvent.
There are some possibilities. Perhaps there was no single buyer that also had the (potentially hundreds of billions of dollars of) liquidity you would need to rescue the bank. Perhaps there were buyers who could come up with that liquidity reasonably quickly (on the order of a week or a month) but certainly not in the 48 hours it took for the bank to be completely vaporized. SVB was intending on raising liquidity from the market, and just the act of announcing their intent to do so triggered the bank run in the first place.
Really it is not so obvious to discern between them at the moment that the crisis is at its worst.
But that's exactly the problem. With higher interest rates, those deposits will be looking for a higher deposit rate. With their assets tied up in low-paying long-term bonds, SVB will not be able to pay that higher rate.
It would only work out "eventually", if the depositors would accept a below-market rate until the bonds matured.
EDIT: This is indeed a solvency issue, not a liquidity issue, as also pointed out below.