Hi there,
I have added a grip to the three Canon bodies I have had over the yrs and I agree with the comment that - IMHO - it definitely gives a more ergonomic and balanced feel to the camera especially with the heavier lens. I have always bought the Canon grips being the grip does form an important part of the overall body and integrates with the electronics of the camera so cannot comment on after market.

I know with my 5D3 there is some integration between battery and camera with a grip and I have heard that third party grips dont support this feature but you would have to take that as hear say and confirm for yourself with the 600D.
To be honest, the main reason I went for a batt grip first up was because at the time I was doing a lot of horse event shooting and struggling to get through the day with two batteries. Hence I gripped up and bought a 3rd battery as the grip will run with just a single battery.
What I can say is with dual batts in a grip I definitely get more shoot time out of the batt's and I believe this due to how the grip manages two batt's. I can see in the menu that the batt's at times have different amounts of shoot in them as the Camera balances load between them. At the time I was shooting a lot of AI servo shots and the 70-200 lens worked hard on the batt's with constant focusing but I generally can get through a day just on the grip now.

Hope that helps.
Tips:
Be mindful of the integration between grip and camera electronics and do your research
Dont mix Canon with after market batteries - either use 2 after market or two canon
Yes - to better feel with the 70-200
Yes to better balance
Note: - my camera was already a weighty best with the 70-200 2.8 as you would know - keep in mind the batt grip with two batts adds to the weight, hence the better balance

I wouldn't have a body with out one myself but this is a personal opinion

Hope that helps

Mike