Firstly, I am going to take the definition of "
complete trapping" to mean "The player's ability to switch the current Pokemon he has out for another on his team is neutralized
for an indefinite amount of time by the influence of the opponent's Pokemon until either said Pokemon has left the field." By this definition, this already does not include Infestation and the rest which if I recall correctly can only trap a Pokemon for at least 4, and at most 7 turns (with an item) i.e. not an indefinite time. For this reason, I am not going to talk about the ban on permanent trapping methods such as Magnet Pull, Thousand Waves etc,
only partial trapping.
I believe that removing all forms of trapping will significantly increase the amount of stall in the meta and make it much more difficult to take out Sturdy Shedinja. Most players use status + partially trapping to take care of the bug. With partially trapping gone, then Shedinja will must likely not get status afflicted to it because a player could just switch into a toxic orb mon. Also, Shedinja could run Magic Coat to stop most forms of status infliction (aside from Sacred Fire/Scald/Steam).
I'd say this is a pretty relevant point and pretty much the focus of my argument here. Of course it is quite easy to say "just use Mold Breaker, Hazards, Rocky Helmet, Entrainment" but in practice these are not really reliable at stopping the bug. Let's take a look at what's wrong with each one of these.
1) Mold Breaker + Hazards
Mold Breaker is not being used much in the current meta for a pretty good reason and that's cause most of its utility that it had in gen 5 is almost entirely gone. Mold Breaker cannot guarantee sleeps anymore because of Safety Goggles, grass mons for Spore, and general status absorbers for Dark Void. Besides, noone is going to keep their Shedinja out if you switch in your Mold Breaker in - meaning you'd probably have to use Pursuit or make your Mold Breaker so threatening that it has the ability to beat anything the opponent can offer on switch in, including Chansey. Of the latter there is only one case I can think of and that is Moldy GengarM and even that has to be wary of Kyurem Black. Besides, anyone with a good play-style will not use Shedinja blindly and allow you to put your Mold Breaker in complete with a comfortable rocking chair and champagne; almost always Shedinja has a switch move which is the first move most players use after Shedinja switches in to scout.
You could use Mold Breaker to set Hazards of course, or use hazards in general. Except, unlike Gen5 all hazards can be removed with relative ease. Almost all the time, Shed is in a team that has ample Defog support. Your best bet would be to use one of Dragon Tail, Circle Throw, Whirlwind or the rest to try and get a lucky phaze to get Shedinja in. Also that's two moves that you've decided to use on your Mold Breaker, further reducing the ability of your Pokemon to have a wide impact.
Or you could use Pursuit. You've already reduced your movepool with a move that solely gets Shed with this move, or the hazard + phaze option, or even all three! Of course, you could try to make it more wide ranging (trust me I have tried), maybe even use it with a Yveltal or something else with Dark Stab but unless you use Spiritomb (which doesn't really have the speed or defenses or offenses to run any form of hazard setting and phazing, walling, or sweeping respectively), you're going to be at risk of an endeavor. So, in this case, its purpose is for killing Shed which it might not even be able to do; it could get taken out before it ever gets the chance to. Then, along with the previous phaze moves, if you're not Spiritomb, you might want to run some form of recovery as well so you have the security of taking at least one Endeavor, or just not dying until Shed is dead. Great movepool you have there, really messes with quite a lot of the tier/s.
There are better mons available for hazard setting, phazing and offensive pressure that aren't limited to only hitting the bug since partial trapping moves are available, but they are not quite able to become powerful enough to finish off entire teams in just 4 turns. Often, the one running the trap move has had to make a difficult choice to choose a stronger move, or limit their coverage to get Shed. The strongest one of them, Magma Storm, has terrible accuracy and is blocked Flash Fire Pokemon, which is not exactly hard to find. Or you know, you could use another Mold Breaker, one for targetting Shedinja and one for other purposes. However it is important to know that with the removal of partial trapping moves, Shedinja starts requiring "the dedication of an entire mon" which doesn't sound really appealing to me. I don't want to have to run a mon that is irrelevant to my team 80% of the time.
2) Rocky Helmet, Leech Seed, Entrainment, Status
These are soft counters to Shedinja at best, because they are easy to find, and easy to avoid. Once again, noone is going to allow you to use Entrainment and kill the Shedinja, so again you could run Pursuit and face the similar movepool arguments I put forth above. If something is gonna try to stop PH with entrainment it's likely gonna be a wall and walls need as much utility as they can get; Pursuit as an option seems pretty poor. A competent player should be able to find out almost all of these methods and prepare for them with dare I say it, relative ease.
Of course, you could argue against the above point by saying that the same could be said about partial trapping moves... so then why ban it if it's predictable and easy to stop?
Therefore, if you remove partial trappers, you would get rid of any PerishTrappers at all. I think it's obvious though that Shedinja is far more common in the tier than Perish Trap is because one's viability greatly outstrips the other. I'd also put Perish Trap with partial trapping something you should be able to prevent or predict via scouting/switch moves. Finally, you
do have a way out after a while. Partial trapping doesn't seem to have the same problems as permanent trapping would and removing them would very likely cause more harm than good.
I am open to any counterarguments of course and the possibility of banning permanent trapping moves, though.