I have an Eisenhower flipper, I believe Tango but bought it years ago so I'm not sure. Getting back into using a flipper again ,but I am unable to get the small portion out for repair. Tried alcohol, acetone and even the lighter as recommended in the repair vids. Nothing works. Any thoughts? Would hate to have to purchase a new one. But the new system does look far superior and easier to repair.

You might check out https://www.youtube.com/watch?v=YQnlsdMsDew. That's a tutorial from Tango magic for flipper repair. However, I offer it with a caveat - I can't get it to work for me. I can't feed the elastic through the channel. Maybe this will work for you.

Because of my issues, I am also interested in people's input for fixing a gravity flipper.

If it's a Mason/Swadling "Next Generation Flipper" (NGF), the small piece doesn't need to be removed. The video shared above looks like it's for Tango's (authorized?/unauthorized?) use of that design, where there's a much larger "channel" around the small piece vs the "groove" on the larger one. Traditional flippers also have a "groove" on the small piece so threading things through generally won't work.
If it's a large channel, fixing per that video should be a no-brainer. If not, the small piece will need to come out - I'd continue to try solvents like acetone or maybe something like Goof-Off, and then gently.... GENTLY.... ***GENTLY*** try to pry the piece off with a flat blade like an exactly knife. More by pushing the blade between the shell and the piece than leveraging the blade to push up, as that can deform the face of the coin.