... transmissions that use a pilot bearing can be a bit wobbly at the input shaft when detached from the accompanying engine.
sifawangiaEV: Can you confirm if your crankshaft has a pilot bearing or not?
I don't think that it has a pilot bearing. I attached a picture of the clutch disc in my starting post so you can see that too for confirmation.
Rather than hard, it's impossible to tell if the transmission input shaft is supported by a pilot bearing (or bushing) in the crankshaft from looking at the clutch disk. Fortunately...It is hard to determine if your gearbox used a pilot bearing based on pictures of the clutch disc. Gotta have pics of those shafts as KiwiME said, short of us doing research into your car's design.
... this image shows that the transmission input shaft is splined to the end, without a plain section to insert into a pilot bearing. A pilot bearing is normally needed in a traditional layshaft transmission, in which the input shaft is very short, carrying only a gear which transfers the drive to the layshaft and a coupling to the mainshaft for the direct "gear" (1:1 ratio). This Mini Cooper is typical of transverse transaxles which are all-indirect, meaning that the input shaft is long, carrying the input-side gear for every ratio; since the shaft is supported at two widely-separated bearings, the clutch disk can be safely cantilevered on the end so no pilot bearing is needed or used.The other end of the coupler that's hidden fits the motor shaft and part that's welded fits the transmission shaft. Here are the pictures:
View attachment 127822
A 2006 Mini Cooper should be the last year of the first generation. If the Wikipedia article is correct, that means that - because it is the Cooper S version - it has a Getrag G285 6-speed transaxle (which was also used in the Ford Focus ST170 and SVT). It is a typical "3-shaft" transverse manual transaxle. This is a photo of the internals of a racing version (with straight-cut gears and dog ring shifting), showing that each of the three shafts (one input, two layshafts each driving the diff's ring gear) is supported by bearings are each extreme end:
The bearings visible (on top as it sits in this photo) are in the end opposite the engine, supported in the main case which has been removed; the bearings at the engine end (the bottom as it sits on the bench here) are in the engine-side case which is visible. The input shaft is the nearest one (without any shift elements on it); the differential housing area is just visible on the far side).
Edit note: replaced image with view from more useful perspective