You can. The school can and by statutory guidance should discourage it, but unless there is a EHCP in place or similar, there are no barriers to withdraw and home educate.
The easiest option depends on the child and what's available in your local area. Some colleges have GCSE programmes, some especially for home educated students with a reduce amount of options, some from Y10 age. There are others that require students to be Y12 age though if your child is struggling, it might be an advantage to have that additional time as generally these are one year courses unless Level 2 BTECs. They do take care of exam entry though which makes things a lot easier.
They don't have to take that many, no one is monitoring that and many schools don't do 10. It would be expensive to do 10 exams privately.
While 5 can get a child onto a college course, it can cause issues after that. I know there are some who do brilliantly from that, but for many - and my own teenager is in this who like yours struggled at that age - their post-18 options include considering further Level 2/3 training. As long as you're both aware of that going in, it can be fine - mine went in intending to do an apprenticeship at that level after as he just needed that time - but some get caught out.
If home educating, you are responsible for finding a centre that takes private candidates, and registering them with all the associated fees. There are HE exam groups around that can help with that.