JMO, just say no to spring renovations. I would tell you that your friend just spreading fert probably won't hurt anything but you really don't know what is needed. The broadleaf, crab will be dead/ dying right about now. The Three things I would do:
1. Soil Test- inexpensive way to budget and plan your fertility for the next couple of years.
2. Renovation- For now, aerate/ seed /starter fert/ (don't know when your buddy put down his lesco product). Maybe wait till next fall.
3. Keep the grass long- mowing height always plays a factor in turning a weed infested lawn into a nice one. I like to keep scraggily lawns like yours on the heigher side for a year or two until I feel the fertility and the stand is where I want it to be. Also, helps crowding out weeds/ use less hebicide.
As I said, these are just things I do when taking over bad properties. No one has the money to bring in tri axles of nice compst/ topsoil so your focus should be correcting any soil problems immediatly. good luck
Jack Burton: Like I told my last wife, I said, "Honey, I never drive faster than I can see, and besides... it's all in the reflexes."