Hiiiiii
Your aux belt is definitely intact, and wiring at the alternator seems good,
Strange about the battery annunicator on the dash not showing on ignition, what other lights are out out of interest? Did any of these show ( on test stage) before this problem came up? As you would expect to see the battery light if the alt was not outputting (as we know its not) but if the LED is defective or its wiring (not sure how it gets this ill have a look at the wiring diagrams ive got and see if there's any info there) I think there were more lights than this out so could be a ground point behind the clocks where they have been moved?
Cant really think of any way to test the alternator without removing it, but its not a massive job really, sort access out and slacken the lower spring bolt on the tensioner ( can sort you the tool I made and its ease) clamp the tensioner up, slip the belt off the alt, undo the bolts and the wiring and its out really,
Id renew the belt on condition, if its near interval it might be a good idea to swap it, the tensioner and idler while youve got access, as it could be curtains if it went
Ill get back to you tomorrow if can think of anything further, the car will run for as long as the battery is above like 11.3 or something from memory, my mates Ibiza went for best part of a day after the aux belt snapped! but you could end up down s**t street if it conked out somewhere so best not to drive it really,
Hope its good,
Pete