Thats not true
On the older system, you could drive away as long as the previous MOT was valid. For example, if you're MOT ran out on 30th August and it failed on testing on 24th August, you could still take it away as you have a valid MOT in place.
However, its now changed and the new MOT takes over the old one.
Having said that, a garage can not stop you from leaving or putting a spare wheel on. It is your choice if you want to drive off providing you know the risks.
Personally, I would chance it and just go to be honest. The chances of being caught are extremely slim