I have an LRHS and it is a very impressive scope for the money, and in the price range, probably the best option for most things. It is definitely not as rugged as NF, if that matters to you. Not all NF are equally rugged, for sure, but they mostly include internal build aspects that the bushy lacks, so as long as you don't plan to abuse it, it should hold up to normal hunting just fine. It is very usable on low power for fast close shots, though the circle definitely does not work anything like the RDS's you are used to.
The SWFA 5-20 probably has better glass for low light use, but it is (a little) bigger, heavier, more expensive, and a fair bit harder to use under field conditions than the LRHS. By that, I mean that the adjustments are quite a bit stiffer and harder to move. Now add gloves and cold, and it is not a scope I would choose for that.
If it were me, I would get the LRHS, and then save for a NF 4-16 ATACR, if I wanted best in class glass for the hunting mission. For everything else you list, the bushy will do fine, though the reticle is a little limiting for games, where the NF options are very good, especially the MIL-C.
eta: sorry, I must have confused a 5-20 comment somewhere along the way. You were talking about the 3-15, which is a very comparable scope to the LRHS in many ways.
The SWFA 3-15 is really a nice scope. Same reticle function and limitations as the LRHS, though obviously the reticle is slightly different. If you haven't looked at the SFP version, you should. For hunting, you will have a real reticle even at low power. For precision or distance you will be at 15X anyway, so your subtensions will be true. For non ELR purposes, as long as you stick to 15 and below, I don't find SFP to be an issue at all. Now, if you're talking about using it with IR and stuff, then that's different, but for daylight applications, I really like the SFP SWFA. Much cheaper too, so if you want to start with an optic an plan to upgrade down the road, there's that.
Been playing with the SWFA and the LRHS this morning because of this thread. Everyone's eyes are different, but the LRHS seems slightly clearer to me than SWFA, when using the same magnification. It is hard to tell, and it may be different for other eyes as well. I definitely prefer the parallax adjustment on the LRHS, and for the extra cost, you are getting a good zero stop built in, plus the illum, which I use occasionally during hunts.
I'm a NF snob, but the LRHS is a really good scope for $750. Mine is on a shorty GAP rifle, which seems to fit together really well. I may very well get another one for a Spartan Precision build I have in the works.