No reason at all why it shouldn't work with a correctly trained category of AFISO
I am going to get jumped on (again) by one or two very predictable individuals here, but there is a huge problem with using a FISO for "controlling" airborne traffic: he is not a CONTROLLER. The job demarcation in ATC is absolute. Controllers get paid a lot more and their unions make sure the separation is maintained.
(indeed, a number of training approaches work on this basis, technically to be conducted only in VMC and for training purposes) in the future.
Is this in the UK? A FISO has no authority to control airborne traffic, so this kind of thing might be just people making effectively blind calls which the FISO has to pretend to not hear.
Obviously, one could make it work.
One way is self-announcement. In the sort of bad weather where people might want to fly an IAP for real, there won't be much traffic, so a working protocol could be established. I was doing that in the USA, in IR training, but that was in VMC so no "clearances" were involved. Once you get away from the idea of a formal clearance, then it is easy, but nobody is going to have the balls for that - anywhere.
Another way is to have a real live ATCO in a hut somewhere; nowhere near the place. He would issue the clearances for the approach. If he has no radar, he would be fairly "cheap"... well you would need several of "him" for 24/7 cover so maybe this would cost about £1M/year, total with the costs of the office etc, but the "hut" could cover a large area, using the same system of multiple antennae as London Information covers a large area. In fact London Info could do it but they would need to employ ATCOs instead of FISOs. I don't think this will happen either - too bold and nobody will spend a single penny for a purely IFR GA service.
Another way would be to use the present LARS service for the approach control. LARS is already funded - mainly to stop GA busting CAS. On bad weather days they are doing very little. On good days they are overloaded and usually almost useless (Basic Service). It would work on real IFR days but unfortunately the system would be brought down to its knees by training flights which are done mostly on nice days.