Was thinking the same. The only possible reason I could see for them not pushing for Indian players might be to maintain that 'premium' of the IPL to keep its value up.
As regard to other players, you could well see them becoming contracted to a specific franchise group as opposed to national boards. They might have a pool of 30 players from different countries that switch between leagues depending on local players regulations.
The branding point is a good one. You look at major sports teams in the US or what football teams in Europe push towards, it's being that global brand (TV money, merchandising, expensive exhibition matches, matches in other countries). You might not be able to see CSK play, but you could see the Cape Town Kings or the Manchester Super Invincibles with their roster of talent. Equally, you might get a 2-week leg of the IPL player in another country as happens with the NFL matches at Wembley or like the Premier League teams were pushing for a 39th game ten years ago.