Is it really possible to get hard water/soap scum stains off glass?

Yes! Soak in a vinegar bath... vinegar desolves mineral build-up & soap scum. I usually use a 50/50 vinegar/water mix, but really, whatever amount you've got is good. let it sit for awhile, an hour or so depending upon build-up, & check to see if it's ready to be scrubbed off. I've had some with a really thick layer of mineral build-up, I let soak overnight.

These stains are from calcium which will dissolve with an acid. Use vinegar, full strength if necessary.

Sometimes they can be etching from aluminum screens...you can use the vinegar first to see if it comes off...and use a paste made from baking soda and water if it doesn't.

Try applying some white vinegar to the stains first, waiting 30 minutes and then apply baking soda and scrub thoroughly. rinse and dry the glass .

And cheap nylon net is great to use for the scrubbing!