I have done a lot of desoldering with solder wick. It is certainly possible to damage boards with solder wick, but I don't find it inevitable.
I don't see any real problems with your board. This won't be suitable for high reliability (space, medical, aerospace) applications, but I don't doubt that it will function and last for years for you. There is a solder splash that should be cleaned up in your follow up photo. A touch of the iron, or just mechanical removal with an exacto knife or equivelant will do it quickly. It is possible that the trace underneath is damaged, but from the photo it doesn't seem likely.
The heat and tap method is good. If you have access to compressed air, heating and blowing out the solder also works well. Don't wear shorts when using either of these techniques, and evaluate your own position on lead safety. Some would find this hazardous to you and the environment. Is not a concern to me.
When desoldering it is easy to overlook keeping the iron tip clean and tinned. It really helps on heat transfer, which then reduces the time heat is applied and reduces the chances of board damage.