Key to using veneer tape is managing the fuzz left from sawing. I used a spokeshave to carefully trim the edges before banding. A spokeshave has a slightly curved blade so you're cutting in the center of the edge and not the edges of the face veneers. Don't overdo it or you may get a slightly concave surface of the finished edge. Another option is to apply a 'sizing' of thinned glue to the edge, let it dry, and then use abrasive to smooth before edgebanding. I don't like this option as I doubt you can avoid rounding over the edges.

There are dedicated veneer edge trimmers or you can just use a plane iron or a wide chisel to trim. Be careful with grain direction.

I doubt any sort of tape will be a long-term solution for drawer slides. Use wax on wood. Poplar should be a suitable and cheaper species for secondary (hidden) applications.