That donut thing is a torsional vibration damper. Sometimes the shafts have them and sometimes not - on the same vehicle too, just depends who the supplier is.
... and now for some out loud thinking - take it for what it's worth.
If you lift the hood, start the engine, put it in first and with the handbrake on, let the clutch up a bit even give it a bit of gas. You should be able to see the engine move a bit, but not excessively. Then do the same in reverse. Another test for some of the engine mounts is to look for a yellow stain under the mount. Some of them are fluid filled and when they rupture, that stuff can come out.
You should move the tires front to back as well, to eliminate a tire that looks good, but has a bad internal belt. If the vibes go away, it could be tires.
How are the front shocks? Bouncy?
Are the brakes down to the metal or beyond? Stuck caliper?
Wheel bearings - how would you test those? Grab the wheel at the top and see if you can push pull the thing and get some slack??
If you grab the CV boots, can you tell if there is grease inside them?
Did you ever say how many miles are on this car?
Just throwing out stuff, perhaps it will spark something else you haven't thought of yet.