What are you seeing wrong with the wheels? Only looking via my phone so can't see a high resolution photo, but look like typical flip-flop clinchers which will be fine with suitable tyres.
Cranks will work to start with as long as your local velodrome isn't too steep (are they 144bcd?) but more typical to ride 170mm or shorter.
Leads through local track boards is a great idea. The bike above still looks okay to me and will get you started.
Last edited by Dalai; 10-25-20 at 03:19 PM.