Do you remember the brand name and model?
You are right though - until you get into some pretty pricey stuff, very often cheap and medium price Bluetooth phones are “all over the map”. The pair I use for myself sound pretty good (at least on MY ears - something to keep in mind, as the shape of one’s ears and ear canal has profound effects). But, they are as you describe, quirky about connections and bizarre connection artifacts.
I (or actually my daughter) just needs to get lucky, like you did. She’ll have them in an environment where theft may be an issue too, so, we want to keep cost as low as possible. Lack of a well known brand name would actually be a plus!
The model number is TXBTHPGRAP for $7.00
it’s a slam dunk.