A lot of this depends on a person's approach to working on these old machines and the furniture. Some people want them to stay in the original condition, some want to restore them to look original, some just want them clean and working. I fall into the "just want them clean and working" category. Where you fit will dictate how you want to proceed.
In your situation assuming the iron is not salvageable, I would start looking for replacements from Singer tables. I have a sister who lives in Minneapolis so I regularly check craigslist for machines for her to pick up for me, and right now there are three different Singer treadle bases, iron only no wood, for sale there, one set as low as $25. If you want to go this route start with craigslist and facebook marketplace. Will a Singer base work? I'm not sure but for $25 I'd be willing to find out, and if it's not an exact fit then maybe I figure out a way to make it work. Once I knew I had a replacement base I would get the glue, clamps, brad nailer, and Howards, and start putting the wood back together.
I started doing this with zero wood working skill, and don't have much more than that now, but it's immensely satisfying to do what my abilities dictate, and when I need help I come to this Board. And finally, I started with zero knowledge about the machines themselves, some mechanical aptitude, and have learned a lot by trial and error and from the many posts and people here willing to help.