I’ve always found the Personal Hotspot consistent. It’s as you describe it but I always assumed it was part of the intentional design: they let you set a persistent password so, to protect you, it switches itself off after a period of inactivity.

I, too, find that I have to toggle it off and back on again for it to show up again on client devices that had used it previously. I figured that Apple has designed it to hide the SSID and deactivate connections after a while.

It was such a strong assumption (which might, of course, be wrong) that I didn’t even bother to Google it to check.