I have some experience with reworking boards. What I found with enameled wire is that it is - for me, at least - very difficult to tell whether or not I got the insulation off of a given area. Also it tangles very easily. I always suspected that it was shorting somewhere and got rid of it. IMO 30 gauge wire-wrap is ideal. It is narrow enough, the insulation (or lack of) is visible, stronger and just far easier to work with. I bought my Falcon030 damaged and redid some trashed traces with excellent results.

And of course if there are lifted pads, good luck filling them if. The solder will not reliably bond to anything. I recommend gently scraping the board mask coating away from the end of the traces and jumpering them directly with fine wire.

When part removal seems rushed and clumsy, it is usually either because someone did not have the patience to adequately heat and remove all of the solder, or the PCB holes were too small for the parts so they were jammed in. De-soldering is far more difficult than soldering.