The 120 day requirement is built into the QTS standards and unless there is an exceptional circumstance where it is not possible to do the 120 days you must complete them.
I can see that you may argue that not being able to find placements may constitute an 'exceptional' circumstance, but if you have a placement and this can be extended in the school beyond the 'normal' placement time so that you complete 120 days then this is what the TDA and OFSTED would expect your provider to do.
This year, as so many schools were closed due to snow, providers did ask for advice on how we deal with this and the 120 day rule. We were told that if the number of days short of the 120 was not significant (no, they didn't specify a number) and if the trainee was seen to be meeting all the standards then we could use our discretion if the trainee was a couple of days short. However, if a trainee was only just meeting the standards or if they were close but not quite meeting the standards then we would have to extend the practice to allow them the full 120 days.
The logic is this; if you do not do the full 120 days and at the end of the placement you were deemed not to meet the standards in full and the provider tried to fail you and not recommend QTS you could (successfully in my view) challenge that fail on the basis that you were not given the full opportunity to show that you could meet the standards. So in some ways this insistence also covers you from being unfairly failed and covers your provider from potential action against a judgement that you disagree with.
Most providers add some extra time in school to cover for the odd couple of days illness (e.g. at Sussex our two placements add up to 126 days in total) and some (including my own place) have some extension time built in that could be used if necessary.
Not finding placement is an issue and in your case has disrupted the training year programme, but if there is time to extend a place so that you can complete the 120 days then I think the provider is right to do this and make such provision. As most PGCE providers finish towards the end of June and school are still open and working into July then while it causes problems for the provider (getting things verified through exam boards) and you - extending the time you are on placement - it is not unreasonable.