I don’t know what size the paddock or turnout is or where you are located, but obviously those things are factors as to what appropriate management is. I know you are hoping to remain anonymous, but it makes a big difference to know some of these details.
Picking manure out of paddocks is an extremely time consuming endeavor in regards to horse keeping. It’s much more work than cleaning a stall if you have to drag a wheelbarrow out in the paddock and walk all over picking up poops (whereas with a stall you can just drive your manure wagon/spreader down the aisle and clean from contained areas). Picking paddocks instead of letting manure break down also results in more manure that needs to be hauled away, and right now that is extremely expensive.
Another point to consider is that in many climates manure breaks down just fine and there is no logical reason to pick paddocks. Horses poop and pee in the “rough” areas of their paddocks and don’t eat from those areas, and horses are not harmed from laying on a manure pile so in most instances there isn’t a hygiene issue from the horse’s perspective. I’m just trying to point out that in many areas, manure is not removed from turnouts for the reason that it would be a lot of labor/expense for quite minimal (if any) benefit to the horse.
For those reasons, I don’t know too many professional barns (even extremely fancy ones) that pick paddocks and fields.
In terms of dragging fields with a harrow, there are reasons to do that to be sure, but that is a very unhygienic practice in terms of parasite management as harrowing spreads manure (and parasite eggs) from the “roughs” into the clean areas.
Yes, removing manure could reduce fly issues, but it might not have as big of an effect as hoped for as typically manure still ends up being piled up or stored nearby until it can be hauled away.
If we are discussing a smaller turnout, like a “run” that is attached to a stall, then in that case I believe it would be very reasonable to request more frequent / thorough cleaning. However, I think it would be wise to keep in mind that likely your boarding situation is what it is and that change may not be forthcoming and you might need to decide if you are comfortable doing the work yourself or if you would rather move to another facility.