Having some experience in MIG and primarily TIG, TIG would have to be my first recommendation. Like someone already said, MIG happens so fast that you have to know what you are doing right off to get a nice weld. Everything about TIG is in your control, heat, amount of filler, etc... are all controlled by the user. This can be a good thing and a bad thing. Inexperienced welders may struggle with applying the correct amount of heat or filler or having an improper torch angle, etc...
I have 0 experience with brazing but it has been used for years by the pros and it obviously works for them. The key to welding is that your joint should ALWAYS be stronger than your material so welding method doesn't really matter provided that the weld joint is done properly.
To test your weld strength weld up a simple T-joint. Then constrain one leg of the joint and apply a load to the other leg until something gives(tube bends significantly, tube breaks, blah blah). As long as the material fails before the weld, you're good to go.
Another way to check is to weld a joint up and then saw it lengthwise to check for penetration.
Notice on this 1.25"x.065 4130 test piece I did a while back that the failure is outside the weld joint(not my best welding but it'll work):
Brazing will likely be the best method for you since TIG welders are very expensive and a oxy-acetylene torch is an extremely valuable shop tool even if you never do a single braze. Heat treating, stress relief, heating parts for press fits, heating stubborn rusted bolts to get them to loosen, and cutting are only a small sample of the many uses of a good torch setup.