Apart the NT coach knowing the skills of a player which is frowned upon if the player has no NT future, there are no downside of having a player called/be in the NT.
Having to ask individually dozens of managers to be able to call a player would be an horrible hassle. Especially as 10 to 20% would never answer and some would answer a week later.
Such a suggestion would be perfect to take NT players hostage and kill part of the NT world.
I disagree, I do not see how it could kill NT world, it only reduces choices for a NT coach. If draft is random, NT have no assurance to have good players, so it's just as it was a bad draft and there will be some player less for NT.
Batum will not play next European Championship, France will play in any case, there will be another player at Batum place.
Ask individually is easy solved, players who wants their player in NT could just contact NT coaches and/or add a button on/off to let a player available or not for NT.
About skills, well, I'm not happy that someone, without asking, knows how my players are. I do not know how it will use this info and as manager I'd have the choice to let know to someone else know how strong are my players. I could have a NT coach in my DIV and he'll know how is my team. It's totally unfair. Or to avoid that I have to buy all players outside the country I'm playing in. It still is unfair.