Mine has been like this since I have bought the car 2 years ago and hasn't really gotten any worse TBH.
I don't feel any vibration throught the steering, that was down to the knackered wishbones.
I haven't got the gear or the bottle to try that though
How much was a replacement diff (and total outlay)? PM if you wish.
Replacement diff was £50 from a "known good" breaker from here ..

Took a couple of hours to take the old one out on my own, and would have been about an hour (with two other very nice OOFers helping) to put the replacement on if we hadn't had a faff with the mounting bolts. Total outlay less than £60 and a few cups of coffee

As to "bottle" .. you have to jack it that high to change the diff, and you have to get under it to change the diff ... and you have to pull and push a fair bit to get the diff out and in .... so actually running the engine whilst lying under it is no more "dangerous" than doing the job IMHO.
Safety is everything I agree, but well supported on decent axle stands and a little thought about what you are doing and the risks involved and it can be made quite a safe procedure IMHO
