Many good and valid points already expressed here. Particularly with "homeguard" who states:
The FIS controller dosn't need to see a screen to give good general advice relevant to the position, altitude and routing passed to them and they do the job well.
Any ATCO worth their licence
only needs accurate information from participating pilots to do that. And it's done well, generally.
To IO540: many ATSUs will have a "discrete" squawk which can be allocated. This allows adjacent radar-equipped units to see instantly exactly
who is talking to that a/c, and if necessary "request co-ordination."