I have a $20 smart fitness watch from Amazon which I use for sleep tracking, though it's a pretty capable watch. Much smaller than an Apple Watch and the battery lasts like 10 days on a charge. It connects to Apple Health and puts the heart rate / fitness / steps / sleep data in there.

If it could do the "make my phone beep" feature I'd probably switch to it as my main watch. But of course that's Apple exclusive proprietary :(