EyeDocPhotog, have you tried dog collars chained to stakes in the ground? OK, OK! I’m being facetious. You have accurately described your challenge however. You want to improve lighting while not directing your extremely active subjects, possibly while trying to avoid making them aware of your activity. That is a tall order!
Where you shoot, and how much you are willing to spend may be factors to consider. Editing tools will definitely be your friend. So will a hot shoe mounted flash.
You could use off-camera flash, but that would work best if you had a small army of assistants with Speedlites on paint poles. Reflectors are like off-camera flash but being larger and more or less flat, they catch the wind. None of this lends itself to children running around and even less to those who don’t want a photo taken so you have to sneak up on them.
Most scrims have the same issues as lights; someone has to position them, or you have to have a really, really big one. We used to live in a house with wall-to-wall, floor-to-ceiling, south facing windows. Mom purchased fibreglass curtains to cover those windows. They let in quite a bit of light and you could see through them. I don’t know if you can still get them but they would have made great scrims! They were extremely large, and porous so wind would have gone through them. With some sewing and a bit of hardware, you could cover a whole backyard. Another material that might work well is the lightest weight white spinnaker cloth you can get. You might be able to get a blown out spinnaker, or a drifter, for next to nothing if you ask around at the yacht clubs. The challenge is that a lot of spinnakers are rainbows or dyed to match the boat’s colours because that sail is not used as much as the rest. The other challenge is that we sail on the Great Lakes where sometimes there is almost no wind, so specialized sails are really light, ocean sailors may use heavier sails because more wind is available. Usually an ocean going sailboat has a shorter mast, too. New York is going to have ocean sailors. A local chandler could get very light material even if they don’t normally carry it, and they could make up a scrim large enough to cover a football field, with enough strength to withstand being caught by wind. The guy that shot the Sears Catalogue cover for years shot the same family from when the kids were little until they were grown. The family happened to be walking on a beach, just when his crew was doing a shoot, and he asked if they would like to join in. They were shooting at noon, in direct sun on a beach. The crew put up a huge scrim, just a level fabric supported by a bunch of stands and some tubing. The photo was good enough to make the cover, and the family was brought back year after year to shoot the next cover with the year’s fashions.
If a giant scrim is too much, how about a Circular Polarizing filter? I would try it in conjunction with hot shoe flash. I would also try an adjustable ND filter set to the minimum darkness, or to maximum light transmission, which ever way you want to look at it. Unfortunately it won’t let most hoods fit, but I have had some really good results shooting tulips with that filter. Don’t laugh! Tulips are a pain to shoot because they are translucent and waxy. They reflect light like crazy and in direct sun they are like little light bombs.
Something else that might work depending on your timing is HDR. My 5D Mk III can be set to make in-camera HDR photos. Press the shutter release once, get three raw photos and a JPEG. It can save the raw files and will do minor alignment, so there are several options for final processing. Your 1Dx should have the same feature. You can probably adjust for 1, 2, or 3 stops between exposures. It eats more memory card, but might work, at least if they stop running.