I don't recall exactly, but the steps go like this:
- A Cabinet member puts the request into the senate.
- The president is immediately replaced by VP.
- The president says, "Nu-uh."
- The VP is immediately replaced by congress.
- Something about 8 cabinet members signing onto the original copalint.
- 2/3's of the House and Senate vote to remove the president.
Since it's all loosey-goosey, both the House and Senate can come up with whatever delaying tactics they want.
And I believe the voting threshhold is higher for the 25 than straight impeachment, which only requires a majority in the House.
But, yes, if it's just about getting rid of the president without criticizing his policies or calling him a crook, I could see it being more palatable as an option if they were forced to make the choice, which they cannot be without defectors.