What do you mean about them wanting you to leave after a short time? That part surprised me.
The rest is very standard in the corporate world, whether accounting or elsewhere. If you want to move up, you usually have to move companies. Promotions are great, but seem to be relatively rare. I think this is unfortunate, since a Senior that is familiar with your work is more valuable than a Senior that's still trying to figure out how to spell the company name…but this still seems to be the standard.
Part of that, though, is because your opportunities are greater if you're looking at all companies than just at your company. Like, in my company and location, there's only one person in the next role up from my current role. For me to get promoted, either she has to quit (and she's been there forever, plans to be there for awhile yet), or our department has to expand to have room for another person in that role…and if/when it does, then I have to manage to be the one picked out of the group of all of us that are in my position (or out of all the accountants in the area, if they open the job up publicly). So, I probably have a maximum of 2 shots at the next level in the next 20 years – one if she quits (before retiring, which would be about 20 years from now), and one if our department expands. Yeah, I can't see the future 20 years away, but slight extrapolation is useful for the point. 😉 In the job local market as a whole, in 6 months, there'd be at least 2 postings for the next level position…so that means about 20 times as many options, and that's in an area with a very low number of professional jobs/positions in general. I'm competing against more people in a publicly-posted job than in something internal, but at least there's more opportunities!
Long explanation to say that your best bet to move up is to move companies, most of the time. If I was you, and my goal was Senior next, I would start looking solely at Senior job postings, and not apply to Staff positions unless your current job is in jeopardy.