If I'm picturing correctly how much spraying it takes to rid an area of flies, that amount of dish soap will have no ill effects on the pool. There's worse, and much more, on the skin of the folks swimming in your pool: soap, perfume, suntan lotion, and, well, "other" stuff. Spray away. That's what filtering and chlorine are for.
Inside the house the soapy spray works well because the flies are those big houseflies and there usually is only one or two that gets in from outside when the doors open and there is nowhere for the flies to go and there's no wind, etc. And soapy water (only a couple of drops of dish detergent is needed) won't hurt anything in the kitchen. Once they get hit with the soapy water spray, it's like flak - it knocks them out of the sky and they're still alive so I put them outside gently (but I suspect they'll still die but at least they're outside the house).
For horseflies, I don't mind killing them. I didn't want to spray the 31.45% muriatic acid for obvious reasons of the spray getting on stuff and in eyes, but I tried the 12.5% bleach which doesn't seem to faze horseflies. The water is so pure right now (straight from the well) that I'd hate to put ANY soap in it and there are only two of us so the load on the pool is more from the flies, dust and pollen that blows in than anything else.
It's lucky for us the load is so low as the equipment hasn't been turned on except to test it but the pool isn't full enough to start it working.
Until then, I've been using this redneck "mechanical vacuum" to get the big bugs on the bottom.

(BTW, that's a pillowcase for a catchall bag -- do you have a good source for fine mesh cloth?)
And I skim the top with the kitchen strainers, which works surprisingly well to catch floating bugs.

Speaking of dust and pollen, I can't run the skimmers yet as the well keeps running out of water so I can only do a few hundred gallons a day.
I've been my own "mechanical skimmer" this entire time of filling the pool - which works reasonably well while the water is 85 degrees.
I even have a "redneck dump method" for the bugs, which is a tall rock in a plastic bucket.
The rock enables the strainer to be tapped on it to dump the bugs out into the bucket without damaging the bucket plastic.
And the rock keeps the mess from blowing back in the pool.

This is just temporary until the water gets to the level of the skimmers.
One of the skimmer flaps is broken - some things you just have to buy the correct replacement part for.
For that, is there a really good pool supply house that does mail order?
I like the other gizmo. It looks like something Spy vs Spy would use on each other!
I still haven't figured out what YOUR gizmo is yet. OMG!
That's for the BIG BOYs!
(whatever it is!)

Obviously it's hand made with PVC (some half inch, maybe, and some three quarter inch, it seems). It has TWO decks, one of which might even swivel to let the measuring cup do something. I can't figure out what the black protrusion is yet though. Maybe it's part of the measuring cup? The front bucket also seems to hold something. Probably powder? I don't see a hose gozinta, so I don't think it's water comes in and mixes with the pre-measured powder and then gozouta into the pool, but it does seem to have a handy polyethylene handle but I don't know what the green flap of rubbery vinylized material does.
Dunno. Maybe that's a garden hose in disguise (the green thing) but what I think the contraption is is some kind of chemical dispenser.
Is it?
I wear out my garden gloves pretty quickly. Nice idea you have for an alternate!
The beauty of those mig-welding gloves is they're all leather and long so they're great for poison oak and they're only about twelve bucks or so.
As such, they don't cost any more than regular leather gardening gloves do but what's great about them is the length along your wrist.
They even sell leather welding sleeves, which I don't have but they would work well for heavy brush work I would think, to protect your arms.
BTW, another piece of expensive crummy pool equipment is the standard pool thermometer.
I tried an electronic thermometer but the water killed it (duh, yeah, what was I thinking).

Why does a pool thermometer need to go from minus 20 degrees F to 120 degrees F anyway?
All I need is about 60 to about 90 degrees (if it's colder, I'm not in the water and it isn't gonna get hotter).
I'm always counting by two and I have to put whiteout on the reading just to see it with my tired old eyes.
And the thermometer inside slips down over time anyway - so you lose a few degrees each time.
There MUST be something better. But what?