Re-evaluate swallowing `persist`/`set` errors in the Persistence Manager

### What is this about?

See some workarounds applied in https://github.com/MetaMask/metamask-extension/pull/39643

that should be undone if we can figure out a good way to just let the error propagate instead of silently failing.

Start by reviewing the workaround in pull request 39643 and then locate the Persistence Manager code that handles persist and set errors. Determine how the errors are currently swallowed and what behavior is expected before removing the workaround; done means errors propagate instead of failing silently.

Written by the indexing model from the issue text.