No, there's no tier benefits. Even Amex Centurion comes with only ITC Culinaire. There are cards like Axis Reserve, IndusInd Crest etc comes with this membership. Crest has some extra perks.
Adding to that, you can do take a shortcut to Club ITC Gold Tier with Axis Vistara Infinite by "tier matching". For Vistara Gold/Platinum members, you get upgraded to Gold tier of Club ITC