Equity Trading in AM - career path?

Sorry if this is wrong sub forum. Wasnt sure whether to post in Trading or AM. Felt people in AM might know more as the role is at an AM firm.

What do traders do at AM firms? Is it purely execution? E.g. PM tells them to buy 69 Amazon shares @$4.2k target price and these traders try to buy under that price.

If any, what would be the "exit opps" for these roles? I have an interview for a quant analyst job supporting the Equity desk at an AM firm. Part of my role will also include research into lowering transaction costs. Can people move to firms like Citadel or Jane Street from these roles? If so what sort of jobs? Quant trading or research?

FYI I am a Math grad with a data science internship at a top tech co.

Also: Is it worth going into this whole quant area/career path or would I be better off trying to transition from trading to investment role at the same AM firm? As I do enjoy Math/Stats/Coding but I want to maximise earning potential so if going down Investment Analyst route is better vs quant research/trader in AM, I will pursue it.

Thanks.