Better Visibility of Log Warnings/Errors Buried by Scrollback

Is your feature request related to a problem? Please describe.
My specific problem is not being alerted of errors when a plugin I'm working on has a syntax error; the error occurs, and then is scrolled off-screen by all of BN's other startup-logs.

This was also experienced by some users in our training class.

Describe the solution you'd like
Sorta like Slack or Discord, showing a clickable "Unread messages Errors/Warnings" in the top right of the log window with a count of each (similar to what we already have when the log window is collapsed).

Describe alternatives you've considered
Adjusting the log level, though log_info is general useful.

Keep the log window closed when started BN, though that adds possibly many clicks to users' workflows.

Might be fixed by #2508; Filter out loading logs, if we go through and tag all our internal logs thoroughly enough to distinguish between plugin loading errors and general startup messages.

Reducing the number of startup messages. This could also probably be fine; the "Loaded native plugin" messages could be downgraded to log_debug.

Review issue #2508 and the existing collapsed log window behavior first, then locate the log window implementation and its startup-message handling. Done means the log window exposes clickable unread warning and error counts, lets users reach those messages, and preserves the existing collapsed-window indicators.