The test failures should be resolved by #54874

I'm not convinced about this approach, as it introduces regressions in the case where the destination is a dense zero matrix.

julia> B = Bidiagonal(ones(200), ones(199), :U);

julia> D = Diagonal(ones(size(B,1)));

julia> @btime mul!($(zeros(size(B))), $B, $D);
  7.338 μs (2 allocations: 1.62 KiB) # nightly
  40.437 μs (2 allocations: 1.62 KiB) # This PR

This is because in this case, the isbanded check introduces an extra loop over the destination. What we miss here is a bandwidth function that looks only at the structure of the matrix, and not the values. Perhaps we should add such a function and make it public as well, but we might want a different name to indicate that unlike isbanded, this only infers the bandwidth from the structure of the matrix.

Perhaps you could put the two versions into a little helper function with two methods, and external packages could hook into that, if they are able to decide for bandedness based on type info.