Getting your Switch’s voice chat to behave a bit more like what you might be used to on other modern platforms is pretty simple, but it requires a little bit of research. You’ll also still need to use the Nintendo Switch Online app to facilitate chatting with folks. Basically, you’ll want to get yourself a pair of headphones (or earbuds) that supports “dual wireless” functionality.
It’s possible you already own a set that does this, but if not you’ll want to try and find a pair that advertises multi-device (sometimes called multipoint), pairing so it can connect to more than one device at a time, as well as dual wireless. The set needs to include a microphone, of course; something like the Corsair HS70 or SteelSeries Arctis 9 should do the trick.
Once you have the right headset, you just have to pair it with your smartphone and the Switch. You’ll then be able to hear your Switch’s audio, in addition to the audio from your phone — which includes voice chatting through the Nintendo Switch Online app. It’s certainly not as elegant a solution as console voice chat, and it requires a bit of legwork (and possibly extra expenses), but it’s functionally the same as what you’ll find on a computer, PlayStation, or Xbox.