How does switching jobs affect career progression

Hello,

I currently have 1 YoE at my current job as an IT support specialist and am trying to change my career into systems administration. I know it can require more YoE to achieve this, and my current manager essentially confirmed there isn't a way for that right now or for the foreseeable future, and won't even let me do some of the projects I've suggested.

My question is if I switched jobs, how does this affect my career progression? I already do some tasks related to jr system administration and since it's a smaller company, I already do more than what my role asks for. Will I be starting off from "zero" or will it have an actual effect when I transition to a sys admin role.

With the current job market, I would be looking for a similar role and maybe even in help desk, which I'm not sure is a positive change.

Thanks!