Let's look at the BL pokemon of these SR weak types and see what else is wrong with them:
Fire
Arcanine: Bad movepool, not particularly good or bad at any role. Howl is its only way of boosting Attack, and is a very slow way of doing it. Hard to switch in during OU matches due to common weaknesses in Water, Rock and Ground, regardless of SR weak.
Blaziken: Not even SR weak, but is outclassed by Infernape besides a few rather specific niche roles.
Charizard: Its only really good moveset is extremely predictable, and is already really risky to try and pull off, this is the BellyZard moveset. In this moveset, Stealth Rock arguably helps it by getting him to 25% health faster, for an easier Salac boost. He has 3 common weaknesses in Water, Electric, and Rock (4x). All of his other movesets are often done much better by Infernape, except in certain situations where a Ground immunity is needed on a team.
Entei: Has a horrendous movepool, and 3 common weaknesses to Rock, Water, and Ground, making it hard to switch him in. Arcanine does what it does, but better, and Arcanine has already been explained.
Houndoom: Has terrible Defense, and average Special Defense. Its 95 base Speed is not enough to compensate for the bad defenses and 4 very common weaknesses in Fighting, Rock, Water, and Ground. This makes it hard to switch in, and very hard to sweep without paralysis on the opponent's fast attackers.
Magmortar: Has the 3 common weaknesses of a pure Fire-type, which aren't helped by a base 67 Defense, and a base 83 Speed is simply not very good. If he had just a bit more Speed, he'd be much closer to OU, regardless of his Stealth Rock weakness thanks to his beefy offenses, but the Speed is what's keeping him down.
Moltres: 3 weaknesses to common attacking types in Water, Electric, and Rock (4x), which really hinder his survivability, regardless of SR. 90 base Speed means that his sweeping ability is somewhat hindered since so many OU pokemon are faster than him, and have super-effective moves on him. The 3 huge weaknesses he has make it hard for him to play defensively as well, especially because of how offensive the metagame has gotten.
Typhlosion: His best moveset (Choice Scarf) is extremely predictable. Choice Specs has very few options, and the Sunny Day sets are screwed by Sandstorm. That, combined with the common weaknesses, and mediocre defenses make people not want to use it enough to make it become OU.
Ice
Abomasnow: Was OU for a very long time. Having 7 weaknesses, with 3 of them being to very common types (Rock, Fighting, 4x Fire), doesn't help it to stay alive at all, and neither does its 60 base Speed. Its actual stats aren't exactly the most impressive out there either.
Articuno: Has one of the worst type combinations in the game, regardless of what SR does to it. It has a lot of trouble walling opponents with only 2 resistances and 1 immunity, with the resistances being to very uncommon types. Add on the weaknesses to Fire, Electric, Steel, and Rock (4x), and that makes it very hard for Arty to switch in even when Stealth Rock isn't up. Taking away Stealth Rock arguably hurts Articuno in a way, because his only other form of real damage besides Ice Beam is gone, because he now has to rely on his teammates having only Spikes/Toxic Spikes up if he wants to utilize Roar very well, and those are learned by few. Articuno also has an awful movepool, with very few offensive moves, and support moves that are learned by several other pokemon that don't have a huge amount of weaknesses, and basically depends completely on its stats. It may be a Garchomp counter without Stealth Rock, but once Garchomps just start running Stone Edge, Articuno's just going to go back down in usage if it went up at all.
Regice: It only has 1 resistance (to Ice), and 3 big weaknesses (Fire, Fighting, Rock) with 1 not-so-huge weakness (Steel). This makes it totally inferior to Blissey at Special Walling, and only useful in niche situations. He doesn't even have a 50% recovery move, which is a huge hindrance to its walling ability already, regardless of SR.
Flying
Aerodactyl: Its 5 weaknesses are a huge hindrance. Coupled with bad defenses, and the advent of the Choice Scarf, it's no longer faster than pretty much everything out there. With Stealth Rock gone, it actually loses one of its strategies with Taunt, and since it's often used as a lead meaning that it doesn't take SR damage all that often, SR being gone may actually be a hindrance to it, since that strategy is what's raising its usage right now.
Crobat: Its type coverage is very bad, and the non-STAB moves it has have only average BP. It's very predictable since it almost always runs Hypnosis, and though it has some potentially awesome move combinations, such as Nasty Plot + Hypnosis, or Hypnosis + Brave Bird, they are illegal, which ruins his usefulness. He has a niche as the fastest sleeper in the game, but that one niche is not enough to bring him up to OU status.
Honchkrow: Though it does have 100 base HP, 52 base defenses a piece is just bad, and makes it really hard for this guy to survive. Having only 71 base Speed, and 3 common weaknesses does not help him survive or sweep any teams. It also has a pretty bad movepool, especially when it's resorting to using Steel Wing on its Choice Band/Scarf set. If it had more Speed, it would most likely be very close to OU, but it's stuck at 71 base Speed, which just isn't fast enough for people to want to use him enough to make him OU.
Ninjask: It has 5 weaknesses, all of which are pretty common, and regardless of SR, most strong neutral attacks KO him anyway (even some resisted ones do). He's also extremely predictable, since the Baton Passing set is his only really good set. He has a lot of trouble trying to sweep at all since he has awful type coverage, with only Bug/Dark/Flying moves to attack with (I'm not counting Dig). There are so many ways to destroy him and stop him from Baton Passing as well. It's very doubtful that he would care much about Stealth Rock, because he's often used to Baton Pass, and then only come back in if he absolutely has to, hence why he's often used as a lead.
Staraptor: Already has lots of trouble surviving attacks, particularly special attacks because of Intimidate. 3 very common weaknesses don't really help this either. People tend to shy away from Staraptor because its most powerful attack has massive recoil damage, which coupled with bad defenses, make it die easier than it already does. Unfortunately, a lot of fast sweepers and Choice Scarfers can outrun it as well, which doesn't help to compensate for the defenses.
Bug
Pinsir: Outclassed by Heracross, and with bad defenses, especially on the special defensive side, it's hard to find a reason to use it in OU, besides Mold Breaker, which simply isn't enough for people to suddenly use it enough to make it OU since it only really helps to beat Weezing.
Shedinja: Has so many other problems it's not even funny. Banning SR would have no effect on this guy's usage, since it still requires way too much support to work properly. Sandstorm/Hail/Toxic Spikes/Spikes/etc. still ruin it, and there's no getting around that unfortunately.
Basically what I'm trying to say from this list, is that banning Stealth Rock is not a guarantee that these pokemon will be used more than they already are, or that they will become OU, because most of them already have huge problems that hinder them a lot more than SR do.