Maybe the tank somehow drains, but then doesn't fill when the bath or the shower is running, but there IS water in the bowl.

Scenario: Go in Bathroom, Turn on Shower. Use toilet. Go to Flush toilet, but NOTHING happens. Toilet lever IS attached, but the tank is empty and won't seem to fill.

I've seen questions where the OPPOSITE happens, but this isn't the case. The toilet fills and flushes fine as long as the SHOWER/Bath Faucet is NOT running.

Oh, and BTW, we've got that constant "whine" in the house that probably has something to do with air in the line as well. Perhaps the issues are related.

Sounds like you have an issue related to volume of water available to the house... maybe?

I'd start by looking at the water main supply valve... be sure it is wide open.

Next, look for a pressure reducing valve... strainer may be clogged and need to be cleaned (see image).

After that, if no obvious issues found I would disassemble the toilet fill valve and clean that... see if that solves the toilet issue.

You do not have an issue with air in the lines... really no such thing unless first filling the main water pipes.

Check these things and let me know what you find, OK?

Mark

Well, I feel silly, but I think our main valve is on top of our water heater in our basement. Does that sound like a likely place?

We seem to have adequate pressure in the rest of the house, but I'll give this a try.

Still have that whine. It's constant. Might not be related, but it's there.

Main water supply shutoff is at the water meter (town/city water) or where the pipe from the well comes into the house (pump/well system).

If the whistle is at the water heater I would suspect issues with hot water more so than cold water, so I would still be looking at the main water shutoff or still looking for the pressure reducing valve (also at the water meter area if present)...

Keep checking and keep me posted, OK?