Independent evidence for the classification of topological 4-manifolds?

After 7 months, over 800 MO views and (as suggested) emails to experts, the answer to the question is "No": other than Freedman's 1982 paper, there is no evidence what-so-ever that topological 4-manifolds are so much simpler (i.e. determined up to homeomorphism by their intersection form) than smooth 4-manifolds. Subsequent research (capped gropes etc) hinge on the key step in the paper - the removal of "gaps" in the "design". While this is a highly unsatisfactory state of affairs for the reasons mentioned in the original question, it is what it is.

There's a somewhat different exposition in Freedman and Quinn's book. I think the main difference is that they use gropes instead of Casson handles. Gropes are made of embedded surfaces instead of singular disks, and introduce some technical simplifications to the proof (they originated with Stan'ko). Richard Stong gave a correction to one of the arguments in the book, although I think it isn't relevant to the proof of the disk theorem.

Maybe these notes (entitled "THE 4 DIMENSIONAL POINCARÉ CONJECTURE") of Danny Calegari are useful for your interests?: https://math.uchicago.edu/~dannyc/courses/poincare_2018/4d_poincare_conjecture_notes.pdf