Tried to remove one GU10 bulb, afterwards lost the entire fixture
Just replace the stupid fixture.  The entire point of the GU10 was to force people to stop using filament bulbs and replace them with Compact Flourescent bulbs.  This was in the days when the cost for CFC was higher than filament and the lifetime was the same as filament so naturally nobody was interested in buying them except California who was struggling with the shortage of generation capacity due to their stupidity in electrical regulation.
Today, with LEDs the Edison-base replacement market -DWARFS- the GU10 market and with LEDs costing the same as filament and the lifetimes of them so much longer, there is no need for goofy sockets to force people to replace lights with LEDs the cost savings and reduced replacement intervals are plenty good enough for that.  The final nail in the GU10 socket was when CA stopped requiring them in new construction.
I haven't seen a new GU10 fixture at Home Depot for quite a while now.  Everything is Edison-base and the vast majority of LED bulbs are Edison-base.  They have a few token GU10's in basic sizes for apartment dwellers stuck with this strange deviation from what has got to be the most foolproof light socket ever designed, but the cost of GU10 bulbs is rising as the demand has been falling and we all know where that vicious circle is headed.
The former owner of the home knew the future of that light in your nook was fixture replacement and left it for you to do.  Take your wife shopping at HD and let her pick out a new fixture and get some good husband brownie points for this one.