oh yeah that was pretty cool on big daddy.
as far as Axl trying to prove they are 'just as good', I don't see it that way
the point of a performance is for the band to enjoy it and for those who hear/see it to enjoy it also
an original will always be "the original"... and while its obvious that comparisions are inevitable, I don't imagine it being a case of "let's put this out to show people we are as good as/better than the original"
Axl obviously cares about the pre-existing GN'R material... he even told the crowed at RIR3 about getting "emotional" when it came to singing/performing/playing them again. I don't see him as trying to put down the GN'R legacy (one he is a huge part of btw). Its natural that a performer/artist will always try to do his best with material he cares about. The new guys also recognize what this material means to the legacy of GN'R and I believe they deserve a lot of respect for getting up for the challenge.
They're like the "new husband" coming to live in the house that was 'built' by the partnership of the previous marrige... now its their home. And though they didn't have anything to do with the dissolution of the previous marrige - they are in the position of having to deal with the fallout. The extended family is like the fans - some may not accept their being there. They may feel like Axl should just give up the house and move out. But the house means too much to Axl. He too build it with his bare hands - his blood, sweat, and tears also. He's never abandoned it... and he's not about to now. The new guys - they're not there to tear it down. They're not there to chip away at the foundation . If anything they're going to do what they can to reinforce it.
I don't believe any performing of the old material in any way different is an effort to 'outdo' the original.... It's just a way of making the house their home and inviting us in to come in and make ourselves feel at home again also.