I wouldn't even think in those terms. While TLs are the owners of their areas and usually the go-to person for most things, when looking at getting promoted to TL there is no requirement of cross training. Will cross training get you to stand out as a team member through interacting with different store teams and leaders? Of course. But a TM cross trained in 10 different workcenters a good TL does not make.
Firstly, you should express interest to your current TLs and ETLs. Most are really good about once they know you want to develop further, they will help you out and give you opportunities to do so (I have had plenty of situations where I make a WTF face when I find out someone is interested in promoting simply because their actions would never suggest it). You should look at taking on tasks with more responsibility and show how you do a good job in your current workcenter first. Your STL will not promote you if they think you cannot lead a team, no matter how quickly you can push that CAF, how many REDcards you get, or how awesome you are with PTM. Is that the way it should be? Maybe, maybe not. But you can learn the basics of your area through training and any TL worth their salt will continue learning in order to become an expert in their area.
Being a TL, in my opinion, is all about being an effective leader and balancing parts of an ETL with parts of a TM. You need to be able to drive results for your ETL, be accountable for your results, and be profitable business wise while being able to still relate to your team and assist with their tasks (hell, sometimes you ARE the team if you're in an ULV or have no hours). When I was promoted to GSTL from GSA, I was given feedback about how great I was as a TM, but what the leadership team saw that I really improved on was being able to relate to the front end team while being able to command their respect when we needed to get to business.
Hope my ramblings will be of some help! 🙂