IMO schools are going to need to adapt their timetabling, possibly by dropping subjects which are currently compulsory to make room for a different compulsory subject. MFL is a more sensible option than tech for a compulsory GCSE IMO. Then there's no reason why someone shouldn't be able to have more guided options so they can take music and drama as their 'free' choices. The fault lies with school and timetabling if that's not currently possible and parents will complain.
If schools timetable 3 (or 4 or 5) core subjects, then have 2 limited option blocks and then 3 or 4 or 5 (depending on the original setup) free choices then everyone should be able to take a rounded set of GCSEs.
You have to do maths, english lang and single science anyway. Some schools may add english lit (1.5 x the teaching gives 2 GCSEs in many schools so that's fair enough) or insist you do double science, or separate sciences. Some schools have a range of core options where you pick your package and that determines how many choices you have left.
Then you can choose French/Spanish/German/whatever and a humanity (which I think should definitely include RE at the very least!).
Then my school had about 6 option blocks and you could choose one subject from each block and most subjects appeared in more than one block. Your choices weren't guaranteed so you had to put second preferences but that did pretty much ensure that you got anything that was very important to you.
Personally I don't consider english, maths, science, music, art, drama and technology to be well rounded. That's a very heavy bias in my view.
Schools will still, as far as I understand it, have the option not to put students in for it. TBH I'd rather see a % pupils put in for it, then % of those getting it, in addition to the current information than just see how many students got it but I do support the idea of being able to see how well the school performs across the board in a range of core subjects than just looking at 5 A*-C GCSE equivalents.