I took off the gaiter and had a look and it all looks nice and newish. No cracks, almost no up/down play in the gear stick. I can even see some white grease. While playing with it, I managed to find the gears again, so I took it for a quick test drive, but just as I got home it went floppy again.
So I popped open the bonnet with an idea to pop a couple of Viagra in the gearbox and took a look at the cable connectors at the gear box. The big round connection at the gear engagement connection seemed a bit wobbly, it can seesaw through about 90 degrees on top of the connection (is that normal?), but I did find the culprit (I think). The blue gear selector cable was pretty much off the ball joint on the selector lever on the gearbox. Whatever secured it when clipped on is knackered. My guess is that when I got the first floppy stick it had fallen off the ball joint, then it must have slipped back on when it got lined up with the ball, but wasn't secure and so could fall off at any point, which it did. So it looks like I need a new whatever it is that holds the connector onto the ball joint (or a new connector).
So, in a nutshell, my Alfa's stick is floppy because it's not grabbing its balls hard enough.
Thanks for all of the input. With a bit of luck I won't need to get under the car again (at least for a few more days).