Warwick is generally thought to be second only to Cambridge now in terms of maths.
You're a bit out of date here. Oxford is not behind Cambridge, in research or teaching. (I have no connection to Oxford) Moreover Warwick is very strong in some areas of maths but has gaps in others. I don't think it is above Imperial or UCL, even if its entrance standards are a teeny bit higher.
(Entrance standards reflect popularity rather than quality. Durham has high entrance grades for maths because Durham is considered a top university but actually Durham is consistently down in the 10-20 position in research league tables for maths.)
I would imagine that without Further Maths, STEP papers (used for Cambridge and Warwick) would be almost impossible.
Warwick accept a grade 1/2 in any STEP paper. The first STEP paper only requires single Maths. Cambridge usually require STEP grades 1+ in the second and third papers, which do require further Maths.
As a poster pointed above, dropping the requirements for further maths would dumb down the courses. I think the focus should instead be on why there is not enough funding for FM at sixth forms, and not more arrangements between colleges about sharing the teaching of FM.
By the way in reality there are relatively few sixth forms which don't offer FM.
The real problem is that students from "weaker" schools don't choose to take it, and are often advised they don't need it. In particular, only around 25% of those taking FM are female, while the gender split of single maths is almost equal. Not taking FM excludes women from top degrees in STEM. Nationally about 40% of maths students are female, so it is not like women don't want to study maths at university. Many of them only find out too late that they should have taken FM to keep open the options of the top half dozen universities.