Repairing non-functioning shifting on an old MTB

I've offered to have a look at a friend's old MTB (90's Raleigh somethingorother) which has trigger shifters, but the rear derailleur isn't functioning, leaving the chain constantly on the smallest sprocket, making the bike somewhat unrideable.

The trigger shifter doesn't appear to do anything, it just rocks about but there's no resistance.

Cable doesn't appear to be detached from the shifter, pulling at it from the downtube exit doesn't result in a broken cable falling out, it seems pretty secure.

Derailleur itself appears to be ok - manually pulling on the cable from the downtube results in a consistent shift through the cassette, however upon release the derailleur settles back on the smallest sprocket.

First thought is the cable is just bollocksed - the bike is in generally poor condition and the cables are somewhat rusty, however don't want to bother going out and buying a new one if the shifter itself is screwed.

Is there anything more I can do to isolate the problem?

Thanks!

Take the shifter off, open it up, realise it's the problem, go buy a new one. And buy a new cable too (inner and outer) unless it comes with one.
If a job's worth doing, etc.

cable is seized in the outer cable near the shifter?

Fit new cables first, you can buy a complete set for £4 in Asda.

I've taken a few different shifters apart with various levels of success. They are so cheap though that if a gt85 spray doesn't work I'd be with Mr. B.

This is one of the few jobs where WD40 is absolutely great. If it's a 'rapidfire' type shifter, as in ratcheted clicky type, then just spray it in the gutty works and hopefully that'll get everything unceased.

If all else failed, you can get a thumby shifter from a local bike initiative/charity for next to nothing, as long as such a thing exists near you.