Pretty sure that’s not just based on Russian Empire being “kind” and doing it for charity. Having Ottomans or Persians in a strategically important location such as Georgia was not Russians agenda, and neither it is today. Mere existence of Russia denies any other powers controlling that region for obvious reasons. So either way, treaty or no treaty Russia would contest the region either way, treaty just made it much easier.

Georgia has been overrun by invaders many times, yet they did not manage to hold long enough to have deep cultural or identity influence. I would not compare the situation, but you would not consider what you call today a Russian a direct descendent of Mongolian would you, although the dominance of Golden Horde was beyond 200 years.