Unfortunately there are no easy solutions.
Yes, those do look like classic FA mats floating in your pond. (Although pics over the internet is a poor means of identification.)
The only fish that truly eats FA are tilapia. They would not breed or thrive in your pond. I would NOT add a grass carp. Usually they prefer any plants you are trying to add compared to eating any algae.
Does your pond ever go dry when your region is in a drought? If so, you could clean out several inches of muck with your tractor - which is probably the nutrient-rich substrate for growing.
Is there any natural "fertilizer" that is reaching your pond? Such as livestock waste, or waterfowl on the pond? Eliminating that source would probably starve the FA at least a little bit.
You said you had trouble establishing lily pads. Usually it is difficult to establish rooted plants when something in the pond is grazing on the plants. Some suspects are turtles, crayfish, and carp. Do you have any of those?
Do you have any big cracked tubs on the property that are now worthless? You could put a couple of inches of gravel in the bottom and then good topsoil in the top and then place it in the pond on a little flat spot at the optimal depth for your species of lily. You might be able to get a good plant started in that fashion.
They do make pond rakes for pulling out the algal mats. Some pond owners on the forum have attached a pool float and a rope. You (or another fit family member) could probably sling it almost to the center of the pond. Perhaps just do a big clean up on the mats before you are holding a family event and want the pond to look nice? You could even do a once/year chemical algaecide treatment if you wanted.
When you do pull the algal matts you need to deposit the waste outside of the watershed basin of the pond. (I assume that is what you did with the tractor bucket loads.) You cannot let the mats decay on the bank, or the nutrients will just re-enter the pond over the next year. You can let them dry on the banks (to get lighter) and then move the FA.
The good news is that you are actually pulling out some nutrients when you remove the algal mats. You won't see much progress at first, but you might start winning the battle after a few years.
P.S. What kind of perch are you catching in the area? If you are catching yellow perch, there is a fair chance that they may thrive and even spawn in your pond. They are fun for grandkids to catch and they are VERY tasty to eat.
You would probably need to lay some cut sapling branches down the pond slopes in 1-5' of water for them to drape eggs in the spring. If you did get a good spawn, then you could try to prevent them from spawning the next year, and just grow one year-class of fish in your pond so you don't get too many fish.
