Applying for 2021 SA with no experience???

I'm an international student at a UK non-target (Russell Group, UK top 20, think Birmingham/Leeds/Southampton...). I'm doing a 4-year degree in economics , and I'll graduate in 2022.

I'm in a very very shitty situation, I haven't done any internships for 2020. My extra curriculars are lacking as well, no leadership role in anything. In fact I have trouble putting together a good cv. My original plans for summer got cancelled because of covid19. The only thing I have is some sort of insight/networking day in consulting at a big 4. I did talk with a director and a couple of senior associates, but I didn't stand out enough for the senior guys to remember my name that's for sure. (I'm an introvert and it's the first time for me to attend these events) I also took some part time jobs last year but they were absolutely irrelevant (customer service at an expo)

Do you think I should just give up applying for the BBs and focus on boutiques or Big 4 internships for 2021? Coming from a non target international with no experience, I don't even aim for IBD as for right now, I just want to get my foot in the door.

My plan is to get whatever relevant internship I can find for 2021 and apply for a top MSF after graduation to break into FO, ideally in London/Hong Kong. Since I'm pretty confident in graduating with a first, I'm looking at Warwick/LSE/Imperial...Do you think this plan is doable? Btw, do you think doing CFA level 1/scoring high on GMAT on top of a high GPA can make up for the lack of experience?