A regular drill is a low-speed, high-torque device. But I really don’t think you want high torque!
Sounds to me like the problem is technique: you want a light touch, and the right bits. I’ve never heard of using a grinding stone for any kind of PCB work. The usual is a small burr or round mill. And a light touch, did I mention that?
As for breaking drill bits in vias: the LAST THING you want is lower speed and higher torque!!! You need high speed, low torque, and very high rigidity. If you are using the Dremel-brand drill stand, there’s your problem: it uses plastic for all the important parts, resulting in an imprecise thing that flexes too much to use for fine drilling. You want a precision metal stand, like what Proxxon offers for its tools. (And those tools have lower run-out (eccentricity) than Dremel’s.) And you need to make sure the board does not shift AT ALL while drilling.
And did I mention a light touch? (I mention this because everything you describe hints to me at too much pressure being used, both in grinding and drilling.)