From my reading steering head bearings should be retorqued after 1000 miles after installing. In theory the first service that is recommended after purchasing the bike this should be done. It is costly to have done by the dealer so it gets passed over. The handlebar shimmy doesn’t get noticed until the 40-60k mile mark and by that time tires are usually blamed for the shake. Retorquing the bearings may help when it reaches that mileage but in most cases it’s too late, the bearings are out of round and it’s best to replace them and retorque after the first 1k and check them every few years after that.
I found the shake bothersome, fatiguing and aggravating...I should not have to constantly have a firm grip on the bars to prevent the low speed wobble.
Having said that, one cannot rule out tires, as many have experienced that changing the tires have eliminated the wobble, as well as making sure the forks were aligned properly, and distributing the weight on the bike. If however you raise the front wheel and slowly go from lock to lock and detect a bit of notchyness then inspect the steering head bearings.
For the record, 99.999% of my riding I have at least one hand on the bars if not both.
. Please be safe