Well, your game is basically a Helicopter clone in essence, fly while the screen scrolls and avoid obstacles, so why not copy the controls from that? Hold down to go down, hold up to go up, release to stay still - or even in most variations, hold to go up, release to go down, which I find makes a more fun and engaging experience but might not be as chill as this game seems to be aiming for. Tapping to get a quick hop of height would be like flappy bird and far too hectic and difficult, but holding to get gradual height then releasing to get gradual decline would feel nice, and so would basically keeping the current scheme, but rather than tapping to start and stop ascent and descent, it would be nice if pressing it down started the ascent/descent and releasing stopped it, so when no keys are held you always fly straight basically.
Helicopter is a game that has had longevity all the way from DOS-based machines to Flash to modern mobile adaptations, it's adored by many, and since this game is designed very much the same in essence, it wouldn't hurt to give some Helicopter implementations a play to see what makes them work for people.