How to get an interview from Quant Trading firms such as Jane Street (UK)?

Hope everyone is having an enjoyable holiday. I've been using this site for a while now and this is my first post - I don't want this to come across in a tone of arrogance. 

I am a final year undergraduate reading for a mathematics degree at a UK target (i.e. Oxbridge) and really want to pursue a career in trading - specifically quant trading. However, I've been unable to land any interviews/pass the CV stage at these types of firms (e.g. Jane Street, etc.) in any year of my degree. Over the past few years, I have applied for all these London based programs/internships/etc. but unfortunately I haven't managed to get any interviews/pass the CV screening stages (except one opportunity, mentioned below). I prep quite hard leading up to application season each year, but then never get to put it into practice.

Some points to clarify:

• I understand that the London offices are usually quite small and thus extremely competitive due to the smaller number of spaces, but I thought I would have been invited to more than one interview in three years

• For personal reasons, it is unrealistic for me to move abroad to Europe and therefore I am not really applying to roles in Amsterdam. I am only applying to roles/firms within the UK.

• My previous summer work experiences have been within research roles at my university with various groups

• I did get an interview for quant trading full-time at a newer/smaller London based firm. I made it past the initial math assessment, coding test, phone interviews, HR interviews, and super day interviews without much issue (feedback from recruiter at this point was extremely positive and was very much indicative that I should get an offer). However, there was a 'seal-the-deal'-type call with a partner during which I was asked an estimation question (number of grains of sand on the Earth) - my first and adjusted answer were still a bit off the real value - and unfortunately ended up getting rejected because of that. The recruiter said that I should just re-apply within the next few years given that I made it that far in the process.

• I did manage to get a full time offer on the sell-side (at a top US bank) for derivatives/exotics trading in the London office. This would start in August 2022. I feel very fortunate to have been offered the sell-side trading job, but I would be lying if I said that the greater salary and cultures of the quant trading firms didn't appeal to me more. 

About my background/skillset:

• I took part in mathematics competitions in secondary school (made it to BMO2 and some later rounds of the science olympiads)

• I did well in my A-Levels and math entrance exams for my course. I am on track to finish with a First Class (don't mean to be arrogant, but just want to make it clear that I am not failing my course)

• I take part in mathematics related extra-curricular societies and competitions (some of which are even sponsored by these types of firms). I am always making an effort to attend the company events on campus and speak to the recruiters to see what they look for in the resume, but it obviously hasn't worked.

• I practice coding on Hackerrank and Leetcode, but am not of the calibre required to apply to high-frequency trading firms for an algorithmic trader role

Finishing questions:

• Are there any tips of what I can do at this point for future application cycles to have a chance at interviewing with these firms and pass the CV screening stage?

• If I accept the sell-side job, is it true that the quant trading firms become unaccessible (because they don't hire laterals from sell-side)? There is always the option to go to a hedge fund from the sell-side, but I'm not sure how that compares to prop trading. (Would someone be able to provide some insight into how a career at a top hedge fund compares with one at a top quant trading firm, in terms of salary/work/etc.?)

The constant rejection is quite demoralising as lots of my friends are able to land tons of interviews from these types of firms (even people doing less quantitative STEM subjects), whereas it seems to be a real struggle for me.

Happy early new year to everyone!