The amount of discomfort (psychological fear of theft or physical nuisance of carrying it) is very likely going to discourage its use. But then you're liable for a fine for not having a helmet, so you choose to not ride the bike.
Just don't require helmets for bike riders over some age.
The idea that use would be discouraged because of that is ridiculous when you consider the alternative: cars or public transport. Both are terrible options. Parking, traffic, etc.
I'm 47 and had a crash recently. How does my age have to do anything if someone suddenly jumps in my lane or a car runs into me?
If anything my healthcare bill will be higher.
The same argument applies to children of course - where do you put your helmet after you've cycled to school? Attach it to the bike loosely and someone will steal it, it's too big to carry in your rucksack all day, and you don't want to have to go back and forth to your locker potentially on the other side of school if you have one at all.
Also if it's locked to the bike you would break the lock and steal the bike. The helmet is the cheaper part. Helmets are often designed so locks can go through them.
Why not? A good bike helmet is several hundred dollars. That's not life-altering money of course, but it adds a decent percentage to the cost of replacing the stolen bike.