In that case, why not just nationalize the banks?

It has been tried, many times and many places. It doesn't work well.
Politicians can't resist interfering in decisions of who gets capital (mainly supporters, also pet projects), and the economy gets sick.
And, surprise surprise, all that misallocated capital leads to lots of defaults, so you STILL have the cost of government bailouts.

Similarly, unlimited deposit guarantees have been tried many times and many places. They don't work well.
There is no level of regulation that can't be gamed, so banks game it.
Some will game it a lot, take too many risks, offer too-high deposit rates to gather assets.
As there is no motive for ANY depositor to care if they are a viable bank, they're the ones that get the deposits...then they blow up.
The blowups are much more frequent and much more costly with unlimited deposit insurance.

Bank failures are a feature, not a bug. They are the feature that prevent this.
Costs to the big creditors (big depositors) are required for them to work: at least a few people have to have a motive to see if they bank is well enough run to hold money.

The main reason for having a central bank is to be the lender of last resort: lend without limit, at high rates, against all good collateral, to stem bank runs.
The Fed's discount window has been very popular this week.
This fixes the problem of liquidity and term mismatch and thus prevents most bank runs, but it doesn't fix insolvency, so there are normally two very different cases to consider.
The SVB case is an interesting borderline.
Like a usual confidence-hiccup bank run, without a motive to panic, the term mismatch would not be a problem and the bank would be more than solvent.
The bonds they hold are solid and will come good, in time: fine if truly held to maturity. The bank was not, in this scenario, insolvent.
But with any motive at all for panic, justified or not, the bank had to liquidate assets at market value and there WAS a solvency problem rather than just a liquidity problem.
Almost Heisenbergian: it's all fine till you look too closely at one metric, then another metric gets too fuzzy.

Jim