On Reach:
They didn't really. The first year of Reach was an absolute atrocity. Full bloom sprint saw a lot of pro's shut the door on competitive Halo.
But, when Halo was turned over to 343 they created zbns game types that were actually pretty good. Also, while the forge colors were dull and ugly there were actually of lot of pretty impressive maps made for competitive by the community that could support multiple game types and were still fun to play/watch.
The competitive playlist was still really good and the majority of the time you matched quality opponents believe it or not. Big team battle had a huge following in Reach and was actually fun to play. If you wanted to play some competitive slayer (no matter how flawed the system) arena was actually pretty good. If 343 had applied the arena system to the MLG playlist, kept it in doubles, and added it in Snipes the game truly would have been close to par with previous Halo's.
Reach had an incredible 1v1 and 2v2 competitive scene and was also still supported by game battles which is what the majority of competitive players spent their time streaming/playing.
Also, the end of the Halo:Reach era was only a year removed from being the flagship fps game on MLG and a lot of people were still holding on to the notion that MLG would pick up the next Halo title so Reach gets a slight reprieve because of that.
On Halo 4:
Halo 4 really was actually pretty good in terms of shooting and the shooting mechanics you said felt "more like Halo". The BR was really good and the DMR was a nice option if you decided to use it.
Where Halo 4 went wayyyy wrong was the introduction of custom load outs, ordinances, and a what in the fuck were you thinking matchmaking.
The thing that I and most people loved about Halo was the balanced competitive nature of Halo. I love the fact that a person on the other team spawns with the same weapon as me, has an equal opportunity to time and pick up power weapons on the map, has an equal opportunity to time and pick up power ups on the map and that was an integral part to the strategy and skill gap in Halo.
In Halo 4 all of a sudden the playing field was no longer level at the start of a match. A player could choose their starting weapon, could have a fkn shot gun as a secondary, and if they camped enough to get an ordinance they could choose a freaking rocket launcher/overshield/sniper you name it. On top of it players now had unlimited sprint as base movement and if that wasn't enough they could simply thruster or jetpack out of a situation after sprinting for a year. Good players figured out how to abuse the system pretty quick and soon enough you had a game that felt nothing like Halo.
One of the major problems people had with Reach was the lack of the 1-50 ranking we had grown accustomed in Halo 2 and Halo 3 so the community was clamoring for a ranking system in Halo 4. We didn't get one. Instead we got a late join, non skill based, shit show of a matchmaking system that was nearly unplayable. Late join was broken as hell as you could join games that were 30 kills deep and you were usually losing 30-5 because 15 other people had quit before you. Late join also ruined the balanced principle of Halo that I mentioned above as the other team most likely possessed all power weapons and you were forced to get raped for 5 minutes because you're battling 12 snipes with your BR. When you did get a match that started 0-0 you most likely won 50-5 and forced kids to quit within minutes because there was no skill based matchmaking. It was totally random and just shoved you into lobbies with people who were usually way worse or way better than you.
343 realized these major errors but it was far too late in the game for anyone to care and take notice. They added a ridiculous hidden skill ranking system that you could view on a website. The numbers were purely artificial as you still matched people who were nowhere near your skill level. They also balanced late join but it is still a feature that is not meant for Halo.
On Halo 5:
I can't comment too much on Halo 5 without seeing its full form.
I know I personally didn't find it too repulsive and at the very least we will finally have a Halo that matches you based on skill. The ranking system of a game is what extends the life of a game. So long as people have something to strive for, a goal, bragging rights if you will... they will play the face off the game to achieve their goal.
I personally don't like the sprint mechanics as they sit. I don't feel like not being able to recharge your shields while sprinting is that much of a penalty. Personally I think the moment you are shot your sprint ability to should be stopped and you either need to turn and fight or find cover.
I also think they need to increase the timer on thrusting. Too many times I had what I felt a well earned cross map kill only to be thwarted on the last shot by someone who was playing out in the open like an idiot and was saved by thrusting out of the final shot behind cover.
Otherwise I liked the shooting and felt like the game with some tweaks can work.