What are other techniques to draw a circle besides using or mimicking a compass tool?
I do not know whether you see a pin and a bit of string as a compass mimicking method, it is likely a very old alternative.

You can do elipses by using two pins and a bit of string.

Or you an use circular items and draw a line around it, just like you draw a straight line along a ruler.
There are special tools which have a series of circular holes to allow to make quickly draw several sizes of circles, if you search on -circle template ruler- you will see several different options. (Here is a link to a commercial site which shows one.)

The best method without any tools is training.
The more you do it, the better you will become.
Sit with a piece of paper in front of you and draw as many circles as you can, look critical and try to improve.

There are several methods of drawing which may work for you.
One is one flowing line, no options to improve once started but a very strong result when done right.
A second one is to build up the circle from tiny fragments. Like for a circle with 10 cm/4" diameter you draw little lines about 1 cm/ 1/2" long and overlap them at least one third, maybe two thirds. With this method you can stop and consider, re-adjust and change the shape while working on it. Even in the best outcome it will not show as strong as the single line approach but I find it an easier way to get near the perfect circle. (I never really get it right out of hand, I prefer the computer to make perfect circles for me.)
Maybe it will help you to have a perfect circle of about the right size near while training yourself to draw one. This can be any simple item which is circular. Samples include coins, saucers, tops or bottoms of glasses or even the rings left by a wet glas and so on.

An other training exercise might be that you use a circle made by one of the mechanical ways (compass tool or drawn around an object, or again the ring left by a glass) and draw your training circle around or inside (or both and several on each 'side'.)

Some people will be able to draw almost perfect circles by hand, without much training, others will never learn it. You have to try to find out how good you are.