I would think the pivot points that allow the seat to move up or down are corroded and won’t move.
spray all the pinned pivots with a good rust removal penetrating oil of your choice (Kroil, Mouse milk, PB Blaster…) and try to get it moving. Putting the springs back on and putting tension on them may help to get them to move up since I would assume the seat suspension is completely compressed and needs to move up and then move down but getting it to move a little bit in either direction is very helpful in getting it t move fully in both directions.
I believe that you could remove the seat and suspension hardware to make it easier to work on and break it loose.