First of all, you are wrong.
The SR-71 would start on a low tank of fuel because of weight considerations...
Yes, that's why you do it...
....for the brakes and in the event of an emergency during or immediately after takeoff.
You don't compromise the entire operation of an aircraft for the sake of brake issues. That's a technology issue to be solved. Two things dictate take off weight - thrust and lift. Mostly lift. Can I get into the air with this airframe carrying this weight? In the case of Blackbird the answer is no. It would never generate the required amount of lift to get off the ground fully fuelled, and its airframe wouldn't let it either as it would leak so much as to be completely impractical. They could have added canards etc. but that would have added a terrible amount of drag. The key to that with Concorde was the wing and no one else got that right in supersonic flight.
This is not a design decision to be taken lightly. The Blackbird from the time it took off until the time it refuelled in the air and got up to an operational altitude was a complete sitting duck. You don't do that because of brake and tyre issues.
Secondly it is not fair to compare the SR-71 and the Concorde at all.
I don't think so in many ways. You look at supersonic aircraft of the time and look at what they were able to do and their aerodynamic thinking - purely from that point of view and what was achieved in those limitations. Carrying passengers, no pressure suits, longevity and reliability of the airframe, lift/drag, no exotic fuel required - the list goes on. Concorde is just infinitely more impressive as an achievement and did it on a daily basis. Most military aircraft can only achieve those speeds and altitudes for short periods of time.
The SR-71 didn't leak because the designers were too stupid to build an airplane that didn't leak. If you flew the Concorde at the speeds that you flew an SR-71 it would melt into a pile of scrap or the fuel would explode.
Concorde expanded hugely by over a foot in flight and at altitude, but it didn't leak nor did it need refuelled in mid-air. There were more exotic alloys that BAC could have used to fly even faster with safety in mind, but, they had to balance that against the cost of the airframe (not a problem at all for the SR-71, obviously), the fact that the plane carried passengers in a pressurised cabin with no pressure suits and also range and therefore speed. Slight difference, and considering the extreme limitations more impressive.
The SR-71 leaked fuel because the airframe got so hot at mach 3+ that the airframe expanded drastically. The SR-71 did not leak fuel once it warmed up.
Unfortunately it compromised it hugely as a reusable supersonic aircraft or an aircraft you could fly on a regular basis. You couldn't use the Blackbird as a basis for further supersonic aircraft and there was nowhere you could go from its initial design limitations. Those limitations were for good reasons but they were still heavy limitations. It needed teams of engineers to count fuel drop leakage and those are the extremes we're talking about.
It also traveled at over 3 times the speed of the Concorde.
Errrr, no it didn't. Unfortunately people look at headline mach 3 figures and don't see the technical overall achievement. Nobody actually knows the top speed of Concorde because it was largely fixed by safety, obviously, and no one tested its structural limitations as far as I'm aware but pushing the boundaries would probably have been mach 2.5. One of the French development aircraft reached 68,000 feet. The Blackbird could also only fly at mach 3 and beyond for limited periods in a mission as there were recorded instances of it simply coming apart.
Slightly off-track, the point I'm making here though is that as a passenger I'm disappointed that nothing has really improved in thirty or forty years. No one has really pushed any boundaries and the world just hasn't got any smaller in that time. The fact that we're having to compare two very different aircraft from decades ago, but supersonic aircraft nonetheless, kind of says it all.