Ugh! I so sick of the GSA vs. GSTL debate! Clearly, GSA duties vary from store to store so arguing about what they do and don't do is pointless. (Apparently GSTL tasks also vary from store to store--no GSTL at my store has ever sorted ESIM). It's true that GSA don't do exactly the same work as GSTLs, but it is a demanding, underpaid role--a statement I could make about many (all?) roles at Target. However, taking on GSA can pay off for someone looking to move up.
To the OP: Your exact approach to pursuing a GSTL position is going to depend on your store's unique culture, but I would recommend that you start by making it clear to leadership that that is where you want to go. If they see you as a good candidate, they will probably show it right away. If you don't get a positive response, see if you can get someone to have an honest conversation with you about what's getting in your way so you know what to work on. During my time as a GSA, I regarded myself as not just support to the cashiers, but to my GSTLs. I felt my job was to take tasks off their plate so they could focus on leadership and to partner with them in regard to leadership to the fullest extent that my own role would allow. I have moved to another area of my store, but I had an excellent relationship with my ETL, GSTLs and other GSAs and they still occasionally come to me for my support or opinions. I didn't love everything about being a GSA (although I did like it quite a bit), but I'm positive throwing myself into it to the best of my ability helped me move up.