You've identified your GCSE Maths grade as a particular issue. Sometimes people leave revision too late, don't prioritise a subject sufficiently, or have other issues with adversely impact their grade. So the grade 4 you initially received doesn't necessarily reflect your true ability.
However, you also got a grade 4 when you retook it. And a grade 4 when you retook it for a second time. So I think we can probably say that grade 4 is an accurate reflection of your mathematical ability. Agreed? A grade 4 is a pass. It's not a good pass, but it's a pass. Do you think we can safely say that maths is not a strength of yours?
A problem we have here is that many of the courses you've said your interested in (Computer Engineering/Systems, IT/Finance, Data Analytics, FinTech) contain a decent amount of maths. What is the appeal of courses with contain a lot of maths content, when maths is not a strength of yours? Are you not worried that even if you manage to secure a place on one of these courses, that you'll always struggle with the maths content?
Your "really want to apply to" list is too optimistic, as I think you realise, but your "Other universities I’m looking at" list is probably fine. You have five entries in your UCAS application, so by all means include one or two "aspirational" choices, but you need at least three which are more realistic.
I always tell people that universities don't tend to be flexible with GCSE grades, and that any application to a university where you don't meet their GCSE requirement is likely to be a wasted one. On this occasion, we
know that such an application will be wasted because you've specifically contacted them to check on their flexibility. Also you're
two grades below their requirement. If you "really want to apply", then pick just one of these - or two at most. They'll result in rejections, but at least you'll still have room in your UCAS application for three more sensible choices.
Well, the adjective "strong" is a little subjective. However, "strong" universities receive more applications, which means that those universities can be more selective. I don't know of (and can't find) any which would accept a GCSE grade 4 in Maths for the sort of degree to which you're hoping to apply.
Unless you were in lockdown whilst studying for all three attempts at your GCSE Maths, saying that that was responsible (even in part) for your grade is unlikely to be convincing.
If you had caring responsibilities for your brothers, then this may make you eligible for contextual offers - you would need to check the criteria which each of your target universities uses (they vary). I came across one recently which defined "caring responsibilities" in such a way that it only applied if you were caring for an adult. That's clearly a tad nonsensical, but it does emphasis the fact that you need to read their rules carefully.
Even if you're not eligible for contextual offers, you should still have your school/college/referee mention your family responsibilities in the reference they write. It'd go in Section 2, which is headed "If applicable, enter any information about extenuating circumstances which may have impacted the applicant's education and achievement".
There are pros and cons to foundation years. They get you to university sooner, studying a subject in which you hopefully have an interest / passion - which is clearly great - but they're very expensive. On top of the tuition fee for the additional year, you likely have to factor-in accommodation costs and general living expenses for the additional year too. Is rare that all of this will be covered by a student loan - and even if it is, you have the extra debt to contend with.
Yes. Computer Engineering is very heavy on maths, and degree courses often demand that you have A level maths. For example,
University of Southampton ("A*AA including mathematics (minimum grade A)"),
University of Greenwich ("A Level: 120 UCAS Tariff points including Mathematics at grade C"), and
University of Birmingham ("AAB to include A level Mathematics").
Also, you really need to be targetting courses / universities and courses where you actually meet the GCSE entry requirements, which I assume would apply to your "Other universities I’m looking at" list above. Saying that they "only require a Grade 5 in Maths" is all well and good, but
you don't have a grade 5 in GCSE Maths. Universities are sometimes flexible with BTEC / A level grades. They're rarely so with GCSE grades.
My advice, given above, would be not to. You're two grades off. Even one grade off is likely to lead to a swift rejection.

A strong personal statement is always good, but it isn't a substitute for the required grades. As I think I've said more than once above, you need to set you sights a little lower - at universities and courses which are happy with a GCSE grade 4 in Maths.