Check for filter updates right after a browser starts

### Issue Details

There are times when new versions of filters are released, and because the extension doesn't always check for updates right after the browser starts, the user may see ads. Checking should be done immediately after the extension is loaded, without delay.

Relevant issue: https://github.com/AdguardTeam/AdguardFilters/issues/186199#issuecomment-2286396956

### Proposed solution

Check for filter updates on every browser start.

### Alternative solution

_No response_