The brazed fillet is bigger than the TIG fillet, so the weaker material is offset by there being more of it.
I don't know enough about the metallurgy to comment on 853 weld vs braze. Reynolds are very approachable so I suggest you drop them an email to get a proper answer. 853 has other benefits beyond the improved weld properties - e.g. higher yield so you can use thinner walls / bigger diameters without buckling to play with frame stiffness and weight.
In terms of fatigue, the fillet brazed joint has potential to be very good (flexibility, fewer stress raisers, less destructive to the parent metal etc) - hopefully compositepro will be along in a bit because I think he has some numbers to elaborate in this area. To balance this, some people quote downsides such as lots of lower level heat trapped in the joint (takes a comparatively long time to cool), annealing etc.
Talking to the man from Weldability SIF, the more basic no.1 or no.101 brazing rods were best for bike frames. Other rods with more nickel content were tougher but not recommended as they could result in brittle joints.
I've got a fillet brazed Kili-Flyer from 1990 which is still in one piece (although not had much use recently).
Also had a 3rd hand "internally brazed" Peugeot road bike I commuted on for ages. It eventually broke near the bottom of the seat tube (but not actually at the joint). That was a mass production technique, built with tightly mitred joints and then rings of brass rod inside the tubes that get melted out in a brazing hearth (so it forms a small fillet inside the joint and nothing on the outside). I cut it up and was amazed how tiny the fillets were (as small as a tig bead) and how it had ever lasted so long!
One thing I really like about working with brass is how repairable it is - no problem at all to completely remove and replace a tube. Trying to do this with TIG wouldn't be very successful (welding on top of a ground down weld). I've got some dropouts which are now in their 3rd frame!