Because the principal isn't paid down, the weight of the future loan repayments is more years away, and they get more affected by interest rate changes.
Exactly. As an example, a typical 30-year fixed will be halfway paid off after 15 years, and so is about half as sensitive to interest rates as a 30-year interest-only.
This should not be a surprise to the management of First Republic.
In theory that's true, but a substantial proportion of all US mortgages are fixed rate and originated in the last few years when interest rates were at record lows, so they're still going to be pretty sensitive to interest rate changes and the bulk of the repayments still pretty far into the future assuming they're structued in the usual way with fixed repayments.
This should not be a surprise to the management of First Republic.