There is a limit to what can be bypassed - the engine still has all the temperature limits for what goes through it.
Read the Blackbird flight manual.
There are total air temperature limits. The F-14 had a total air temperature limit of 388C, measured on the fuselage before the inlet.
For the SR 71 the maximum compressor inlet temperature, measured just after the inlet spike, is 417C. That’s hot. Aluminum softens about there. The faster you go, the higher that temperature gets as a result of stagnation. It’s physics, you can’t get around it by bypassing the air because the air that comes into the engine inlet at about Mach 3.3 will reach that temperature. If the air is really cold that day, say 10 or 20 C below normal, then yeah you can get another maybe .1 Mach out of the airplane and stay under that CIT.
Similarly, there are limits on the exhaust gas temperature. Those are measured just after the turbine. Again, it’s physics, the faster you go the more that temperature goes up. The compressor inlet temperature goes up and the temperature goes up through the entire gas path through the engine. Eventually, you reach a materials limit.
The whole engine bypass system in the J58 served to keep the compression ratio down to a workable limit. The air is compressed by the spike in the inlet, then again in the compressor section, and then again as it’s burned in the combustion section. At high speed, that compression ratio keeps going up because of all of the inlet compression. But there is a limit to how much the air can be compressed. It’s kind of a mechanical thing, and the J58 itself had a relatively low compression ratio compared to contemporaries like the J 79. Very high speed, the air would simply choke out the engine because of that compression ratio, so by bypassing a significant amount of air around the normal gas path, you were able to lower the compression ratio through the compressor, combustor, and turbine. Then that bypass air was burned and heated in the after burner. which added thrust, while keeping the compression ratio internal to the engine, low enough that it would still run.
The GE F110 in the F14 had a higher compression ratio, and more thrust than the TF 30. But the airframe had the same top speed with both motors. Because the faster you went, the higher compression ratio in that GE engine, started to limit the amount of air it could move, while the TF-30, with its slightly lower compression ratio, actually continued to increase thrust as the airplane went faster. The fastest I’ve ever been in the F-14 was in a TF 30 equipped airplane. Less thrust at sea level, and at low speed, but high speed, pretty good thrust.
Look, heat is a big deal in this discussion - if the outside of the airplane was heated to over 600°F at that speed, the same thing is happening inside the engine. The outside of the airplane is heated because of this phenomenon, the stagnation temperature/total air temperature, and it raises the temperature of the whole airplane. The SR 71 is famous for this, being made of titanium alloy so that it could withstand that heat, and famously the airplane stretches in length by about a foot when it’s a cruise speed.
But the same heating that happens on the outside of the airplane is happening throughout the engine inlet, and the engine itself. And eventually reach a temperature limit of materials. What makes the airplane so impressive, is that they were able to go significantly faster than anything else built.
Ever.
https://www.sr-71.org/blackbird/manual/