Should not it be as simple as unplugging 2 or more RAID drives, and then
plugging them into a new machine, going to the RAID controller and adding
them as an "existing set".
It is that simple if you have a compatible RAID controller.
I have done it in the past when moving three 1TB drives (RAID 5) from a 3ware 9650 to a 3ware 9750. However, I called the manufacturer before I did that to ask if the on-disk format was the same.
It doesn't seem to be the case, the little bit of information I have found
online advises a lot of caution or to avoid doing it all.
First of all: Caution is always good when handling data. As is having backups.
The real 'problem' is that a 3ware RAID disk might use a different on-disk format than a Adaptec RAID card. Or even different from a different branch of RAID cards from the same manufacturer. Thus you should always check if you can transfer an array from one RAID card to another.
(Do not just try it. If you are really unlucky, it is mostly compatible and you will run into error later. Ask the manufacturer or check the manual. If it does not explicitly mention that the RAID cards are compatible, then assume that they are not. In that case, recreate an array on the new card and restore from backup).
Since you wrote "going to the RAID controller" I assumed a hardware RAID controller. Not software RAID (or pure software RAID or software RAID with BIOS support, e.g. the thing called 'fake RAID').
Just saw your comment confirming HW RAID.