I also struggled with this transition a bit so here are some of the things I learned that help me get to L5:
1/ The key thing for getting to L5 is to take on large problems (usually 3-6 month long efforts), figure out how to solve them, and ship them successfully. You can do this by leading a small group of people (2-4 engs), but also by working as an individual, so I don't think leadership skills in that sense are required for L5.
2/ Usually managers are quite good at coaching for this transition (things get harder going from L5 => L6 and further on). Try to find a team that is seeing some growth and with a manager you enjoy talking to. Managers with some experience (2-3 years) are better at figuring out what gaps you need to fill to get promoted.
3/ Also try to figure out what type of team you will be successful in. For introverted people often product teams are not a great fit as they require tons of coordination to ship anything: you might have to work with a designer, a content strategist, a product manager, and a data scientist for a typical L5-scope project. Infrastructure / backend teams can offer better opportunities for people who want to grow as individual contributors. I know many senior engineers (L6+) in infra teams who aren't tech leads leading large teams -- they are simply strong engineers who can solve complex problems on their own.