If you’re looking for a remote control for your Android phone, your best bet is a Wear OS smartwatch. Try to match the brand of your watch to the brand of your phone if you can, because some features might work better if they’re from the same brand, like with Samsung and their Galaxy Watch.
Hmm, the two points you’re after – not too expensive but also decent quality – can be tricky to find.
Many budget watches can last a long time on a single charge but you might lose older health data (which you said isn’t a priority for you). Also, some cheaper models might stop syncing with your phone after a few days, so just be cautious about that.
Honestly, it sounds like the most basic, low-cost Chinese watches could work for you. But I would still suggest reading reviews from trusted sources because lots of them sound good on paper but are poor quality in real life. I’d guess at least 80% of the cheap ones end up being e-waste.
That said, there are some models that give a decent value for the price. I’ve written reviews on a few thinner models here. (I’m personally after a thin, discrete smartwatch, and it’s tough to find one, especially from well-known brands. Most are either too bulky or too cheap.)