More flexibility in low light performance and range both wide and tele with option #1. On the 600d you would have had to find a lens that is 15-65mm to match the same focal length of 24-105mm on 6D. DOF of f/4 on 6D is also compared to f/2.5 on 600D. The boost in ISO performance and overall look of the images is also better with full frame, but I wouldn't spend that kind of money for street shots if I could barely afford it, you should make the bigger purchases if you know your finances can take the hit or if it will be an investment that pays for itself short term. But of course, in the end, you still buy what you want because it's your money.
Certainly FF has advantages of more control of depth of field (DOF) ie by definition of having a larger sensor, a shallower DOF is easier possible - with the same aperture. And FF has the advantage of generally lower noise, so in that regard the 6D would be a good option.
However there is a place for APS-C (eg 600D or 60D) too. Firstly there are many great EF-S lenses, eg the 15-85mm f/3.5-5.6 which covers 24-135mm in FF format. True, this is a slowish lens. The 17-55mm f/2.8 is a faster lens. Though for truly shallow DOF, a dedicated fast prime (eg f/1.2 - f/2) is more suitable.
UWA, there are so many great lens options - from various manufacturers, many of which are sharp corner to corner, contesting the best L UWAs. Eg the Canon 10-22mm, Sigma 10-20mm, Sigma 8-16mm.
The 7D's AF is great for street photography (19 cross type AF points, well spread across the composition). Most EF-S lenses are less expensive than their FF 'equivalent'. No doubt if money is no barrier, then go for FF- and get a 6D (or even a 5DmkIII or 1DX). But generally people don't choose the 5DmkIII or 1DX because of financial limitations - and needing to keep sufficient funds for decent glass.