Good luck with your decision of "NO SNAILS". I have little ones and at one point had put two Apple snails in from the aquarium but they were impossible to see as they were dark brown against the black of the liner and gunk of the bottom. I found them eventually DEAD mid fall. The snails come in on plants you purchase. They don't cause any problem for me at all but suspect that the ones in the Skippy die back and or get pumped out when we empty it for winter. But somehow, come spring I see snails here or there. No problem with them but I would NOT add them in intentionally. Yah they eat algae on the surfaces but not enough to make any difference.
NOW in my opinion having had a pond about 12 years====and every pond owner has their way of doing things that works for them,,,,I have a waterfall with 4 overhanging drops/steps. When I'm struggling with gunky water, I put filter media in a basket that fits under two drops in the waterfall and add another sheet of filter in my skimmer box. I have to commit to cleaning them frequently but I get it mechanically cleared up. Anywhere you have running water, try to filter it more than normal. IF all else fails, use a chemical (don't shoot me) but be very careful in dosing as you can easily kill your fish. Another thing if you do use an algaecide, be aware that you need lots of filtering then too as all those dead "pea soup" fibers are now laying on the bottom of your pond.
I can see in mine that string algae is beginning and it's way early for cover from the water lilies. So I might just sneak a little algaecide in, but don't tell anybody. This pond about 9 years ago had SA so bad (because I didn't do anything about it) that the fish had made little trails in it to surface. Eventually we both got in the pond and would reach down as far as we could and swish our hands around grabbing hands full and then bring it up, wring it out and toss the clumps into a bucket on the shore. Two of us worked about an hour that way. That is when I learned that I had to stay on top of any algae problem and resort to the algaecide before it got out of hand.