Over the past days I could spend few minutes with the UT-106, study the problem and solutions.
1. Magnet.
The magnet is stuck to a metallic spacer that is supposed to be glued to the antenna base. I re-glued it and reapplied the adhesive paper: it seems it holds better now.
2. Length.
Some reviewers report that the UT-106 is not cut for HAM radio bands, but resonates higher (so it is too short). I don't have a proper connector adapter to measure SWR but I trusted those reviews and applied what I originally wanted to do: cut it to a 1/4 wave on UHF just below the upper coil. The lower coil is still there, but the resulting antenna is stiffer and does not wobble freely, which means it does not pull the base all the time. As for SWR, 3 metres of RG-174 cable introduces enough losses that the transceiver won't notice an antenna mismatch.
3. Testing.
First on-air test was positive as I could hit the repeater from the underground garage, which is not possible with the UV-82 stock antenna.
Second test was on the road. I could see the antenna through the glass roof and it is stable even on bumpy asphalt at 60 km/h. Signal reports from the usual friends were positive as they know how I got through every morning with the rubber duckie.
Third test at high speed. The antenna is positioned above my head and I can see it through the glass roof. My car has no flat metallic surface, so the small base does not touch 100%. At 117 km/h the antenna begins to vibrate: probably a bit faster and something bad happens. I can try to fix this with a small spacer that reduces the front-back empty space, but that's low priority considering that I rarely drive out of town and speak on the radio.
Fourth test at "DX" distance. This surprised me most. I could hit my usual repeater from far away, through a valley where I could do the same with a 20W mobile RTX and a better performing antenna (theoretically).
So after all I will keep this antenna and enjoy mobile HAMming. But I had to modify it to suit my needs and safety requirements.
Now I do want to check the SWR.