The little drill is a nice little device. Only problem is when the battery dies you either replace it or have to wait to charge it. I gave up battery anything years ago as the batteries were either toast, or were never ready in the occasional instance that I needed the item. My old trusty 30+ year old Craftsman plug in drill works every time and at most I need a little extension cord.
As to the flex shaft, it has nothing to due with the mounting. If you are using tiny drills, the length of the shaft is the problem. That length with the weight/power of the motor at the other end will bounce it the tip jams at all. It is great for doing stuff that needs power or has to run a fairly long time. With tiny drill bits however it is going to break them at some point as there is too much mass and power going on. In addition, you are going to constantly fight the flex shaft's weight and pull.
I did this:
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Remember that clear plastic is rather brittle as well, and a small drill can grab and pull as it breaks through the back side. I drilled all these holes with one very thin bit and did not break the bit. I can promise you that a flex shaft would have broken the bit.
With this:
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While the Dremel has some weight, it stays constant, and the power cord doesn't put any real pull or tension on it. Also when running slower with little load it does not heat up. The speed control sets a constant speed that once you have drilled a few holes, becomes real easy to know how to handle and hold it to feed the bit and not break it. I have used this set up for over 30 years and while I have used some quality jewelry flex shaft units in the past, I wouldn't trust them with any tiny bits.
For sanding, grinding, cutting, and drilling larger holes and such a flex shaft will work well, but I still prefer this setup for the precise control it gives.