Usually when the gear starts popping out the syncromesh is starting to wear on that gear.
You might get away with replacing the fluid for a little bit. But ultimately the gears are worn.
Unless there is significant movement in the engine mounts it shouldn't pop out of gear.
I've just watched the video and there is a fair amount of slack in the bushes looking at that gear stick. However mines been like that on all the boxes and gearboxes I've had (3 total) mainly because I've not replaced the top hat bushes. But none of the gearboxes drop out of gear.
I couldn't listen to the sound but generally grinding just shows how much wear on the sync there is.
Your options are :-
1) Remove gearbox, Strip down and replace worn parts = Expensive
2) Remove gearbox, Replace with reconditioned gearbox = Moderate cost
3) Remove gearbox, Replace with second hand gearbox = Cheapest
4) Live with it and avoid third = Free
When doing the above it maybe worth purchasing a new clutch and slave cylinder. Save on labour if you are paying for it.
Or do what I have done in the past if you can't be arsed.
Get a quote for a clutch from Mister clutch or other garages... They need to remove the gearbox to replace the clutch. When they refit the gearbox get them a replacement to fit. Avoiding more expensive labor.
Alfa 166 2.0 Twinnie - LPG and a few trick performance parts