US applicant to econ in UK - supercurriculars?

I am from the US and planning to apply to competitive UK unis for 2027 entry like LSE and UCL. I have good grades and will be fine with the academic entry requirements. What are some supercurriculars you would recommend doing? I have a lot of passion for the subject but I'm just worried I won't have enough evidence of it to show. I am interested more in the social aspects of econ, like how it can be used in social and public policy.
- I am already trying out essay competitions but they are really competitive so I don't think I will win any.
- The biggest thing I have going for me is a project I started to improve my state's high school economics curriculum in which I will present to my state government, but idk how good that is for UK unis since it isn't exactly economics.
- I'm also planning to take multivariable calculus next year (my final year of high school) at a local university, and I am looking a bit into econometrics and stat in my own time right now.
- I have been doing reading in economics and watching lectures + talks by economists
- I want to do research, but I'm scared that cold-emailing professors will fail. Any advice from someone who cold emailed successfully?
Do I have any chances applying if I keep doing what I am doing? What else can I do?

It's not about winning essay competitions (nice bonus if you do but not essential), nor necessarily making a big splash. The point is they want to see you've engaged with the subject beyond the school curriculum, and they want to see what your own reflections on what you've done are, and what analyses you presented in those activities.
Even if you did win an essay competition for example, if you just wrote that you won the essay competition that doesn't really make a very compelling personal statement entry anyway. What they want to see is you discuss how you approached the topic, how you researched and explored the area and then constructed your argument, and possibly also your overall reflections on that whole experience.
Note you can well get into said universities with just wider reading. So the other bits are just other opportunities, but you can demonstrate the appropriate analytical skills and exploration beyond the curriculum just with wider reading. They are well aware not every student has the privilege or connections to e.g. study college level classes, take part in research with acadmics first hand, or have the support to pursue e.g. competitions and olympiads etc. What they do expect all students to be able to do is read around their subject and (this is the important part) thoughtfully and analytically discuss what they've read in their personal statement.
Basically I think you might be getting too caught up in a mindset of needing to have a great list of grand achievements which can just be presented as showing that you're really a special star that is unique among all other applicants, without necessarily much critical analysis or reflection involved. But what UK universities really just want to see that you're a good student who thinks about what you read and do - and they know you're just applying to undergraduate study so they're not expecting you to reinvent the wheel or be doing masters level research or first year undergraduate level classes already (the point is that you're going there to do that stuff!)
That said, LSE and UCL are pretty economic theory focused with a strong quanitative emphasis as I understand. It sounds like you're more interested in the broader implications of economics so are you certain that a course in e.g. politics, political economy, maybe PPE, or the ISPP course at LSE may not be a better fit for your interests?