Yes, a 4x4 that is 20’ long will work. A 4x4 that is 12’ long will support about 4,000 lbs. each (depending on the species and grade). A 16’ 4x4 will support about half that and a 20’ 4x4 will support almost 1,000 each, if its horizontally braced about half way up. (Make the end look like the letter “A”.)
If you get over 12’, I’d worry about the swing “racking” if a couple of 300 lbs. guys start swinging sideways. The metal braces look strong enough, (those are fillet welds which will hold about 1,000 lbs. per inch), but the entire swing set will want to “parallelogram” on you (with enough force). And that “enough force” could come from a couple of drunk buddies celebrating your birthday.
When you exceed the “norm”, you had better be extra safe. I think attorneys call that “an attractive nuisance”.
I’m glad you added that last sentence, “you don’t want to compromise safety”.