The jaws appear to be closing properly. To me, the problem seems to be, they don’t have enough thickness in the gripping surfaces to override the rotation of the bit. The frustrating part is when the now loose bit, goes up the hollow shaft of the tool. I think a large part of the problem is the hollow shaft allowing the drill bit to migrate when drilling pressure is applied. I am going to see if a piece of brass rod filling the shaft makes any difference. Now, I have to find that piece of rod.
Although the collet set I bought grips the smallest bit (#80) I have, I notice there’s a little bit of eccentricity (wobble). You said you use a Dremel adjustable chuck. Is there any eccentricity with that? Also, you said it will grip a bit as small as #80.
The only Dremel adjustable chuck I’ve found won’t grip below 1/32 in. which is equivalent to a #68 bit. I called Dremel and asked if they had a chuck that would grip down to a #80 bit; they said no. Is the chuck you have one Dremel offered some time ago and is out of production?
Make sure with the Starett’s you do not have debris in the slots of the chucks. Clean them by folding a small strip in half, then insert into the chuck slots, move back a forth a few time will clean them.
With a #80 drill any debris will not let the jaws close to hold the drill.