This is how it works, because the rules are set this way.. Why bother popularizing any sport, when you can print the documents and "hire" a few people to secure the budget. This is the most important thing. Being an Olympic sport gets you Government funding.
You say a dying sport. Well that can be said for half of the Olympic sports. Do you think more than 10 nations will still care about wrestling, if it's not an Olympic sport, or archery, or taekwondo, or canoeing, or diving and i can go on. Heck, some of them can't even get 10 competitive nations now.
You save sports from the bottom -> up. Interest is not given, it's built. But that is work, hard work! You have to start from the children, from the schools, you have to have clubs, town clubs, regional clubs, coaches, staff, facilities. But no one wants to wait 10 years for this. It's much easier to get 3-4 Russians etc. to have some results, get the funding and if something else pops, it's a bonus.
I have said it numerous times here, but switching nations should be made extremely hard. To represent you have to be born there, or both parents have to be from the country, or you have be living there for 15 years (even though in modern free times, this not very relevant) or take a lengthy non-compete clause (4 years at minimum like a doping ban).
And i know this would never happen. Because all these sports will shoot themselves in the foot, because the "quality" and number of nations will drop. Or they will bring some EU law, about work rights, human rights etc. Like almost every football club has 10/11 foreign players in their line-ups.