Good point about breaking howards iterations into howardssparse by using Tan improvement approach in reverse. I should try do this, was a nice bit of code you wrote 
VFI Toolkit does transitions based on shooting algorithms. When BM2022 say ‘using a fixed point method’, I believe they meant a Newton method (so update based on the Jacobian, that is, based on derivatives). As you say, they did Type-I Anderson acceleration which is like using the average of the Jacobian over the last few iterations.
At some point I want to add the Newton method, and Anderson acceleration version, for transition paths to the toolkit. But is not an current priority for me as the shooting works fine in my current research papers.