This has the potential to be one of the most moot topics that I have ever wasted my time correcting. So let me give it a try and see how much of my time I can waste.
First of all, the opening statement (title included) and the topic conflict greatly. I mean by the sheer logic of the statements that open this argument, you are saying that if the WWE closed and TNA attempted to give wrestlers (who wrestle for a living) jobs, then they would be WWE's dumping ground? By that sheerly flawed belief, then would WWE not be WCW's, The USWA's, the AWA's, Smokey Mountain Wrestling's, and ECW's dumping ground?
Secondly, I would assume that you are hinting that TNA seems to pick up guys who get released from the WWE. Well I don't know about u, but I doubt that they could acquire them while they are still under contract. And are these guys supposed to NOT work anymore because they get released? And with the sheer number of releases that the WWE has ad over the years, TNA has picked up like what, less than a couple dozen at most? In reality, that's only a fraction of the guys who WWE has let go. And by THAT logic, you are also calling WWE a dumping ground as I believe that such men as Mick Foley, Triple H, Farooq, Stone Cold Steve Austin, Edge, The Undertaker, Diesel, Psycho Sid, and and endless host of others were at one time or another released by other companies before being picked up by McMahon and finding success in the WWF/WWE.
So to answer your question, yes, TNA should go all out to sign talent if WWE closed it doors. Because the fact of the matter is this. Sometimes, creative has nothing for some guys or wants to replace a wrestler. And when this wrestler gets released, a booker might see something in this wrestler that the other company might not have seen. or they might have a storyline that they said "you know, if I would have had this guy I could have pulled it off", and decided that now that the guy is free that they are worth signing. Or a guy just may still be able to get cheers and pops from fans and can be used for cheap popularity. I mean with the exceptions of probably Angle, Booker T, and Christian, I don't see TNA breaking the bank to sign a guy who has a mortgage, prolly some kids, a wife, and needs a paycheck. If anything, TNA can usually get these guys for less than what WWE paid for them.
So it's win-win as a guy gets a chance to start over and the old company gets to cut weight. I mean would you piss on McDonald's if the hired a guy who used to work for Burger King? Hell no. Why? Because experience does matter ya know. It matters in real life, and it sure as hell matters in wrestling.