Yes - looking behind you (briefly) is a perfectly good way to find out what's there.
The dangers of not-looking-behind are much higher than a brief interruption of looking forward.
You're right that your ears can tell you some things, but not if the vehicle is quiet. I've had a full sized diesel-electric train sneak up to within 20 metres unheard by me. Modern electric cars can be so quiet that only the tyres are making noise, and bicycles are always quiet.
Mirrors can be handy too - I use one on my bents because I cannot look behind from a reclined seating position. I also have a barend mirror on my dropbar road bikes because its convenient when you're used to them. I've never liked temple mirrors or helmet mirrors though.
The Garmin radar is technically brilliant, but it doesn't show everything and you can't trust that the way is clear because the radar says so.
Instead I suggest you consciously follow through some steps.
Start 10~15 seconds before your manoever.
Look forward at what might be an obstacle in the next few seconds. Parked cars, pedestrians, moving vehicles and bikes are worthy of your attention.
Unweight your offside hand (the one nearest the centerline of the road) from the bars and make sure your nearside hand (nearest the footpath) has a good grip.
Then twist your shoulders so that in one movement you can see back over/beside your shoulder. Ideally you want line-of-sight from both eyes directly behind you, to help gauge distance and speed.
Revert to forward facing and either act or slow/stop the bike based on the conditions
Have a second glance just before you begin your manoever (not after beginning) You never know what's changed, or what you might have missed first glance.
Be safe out there. If you're ever feeling unsafe, change things until you are. Even if that means stopping and walking.
Your legal requirements might be different, but for me, the road vehicle "doing a manoever" has to make sure the way is clear. Since a bike is a road vehicle for the purposes of this, a cyclist cannot blithely cross traffic lanes without checking first. Don't be the cause/trigger for an accident.