Does TBCC (NASA) need such active cooling? I know the Chinese TRRE does not and that is a purposeful choice.
What I have read is "TRRE发动机关键技术分析及推进性能探索研究/Analysis of Key Technologies and Propulsion Performance Research of TRRE Engine" on journal "推进技术/JOURNAL OF PROPULSION TECHNOLOGY". 2nd edition of 2017, volume 38.
In the article the author considers:
Due to the fact that SABRE uses liquid hydrogen and liquid Helium as fuel and (cooling) medium, the system is (too) complicated (to) lack robustness. It is not suitable for working condition that is time critical. Also due to its upper limit of air breathing mode is only March 5.5, its overall specific impulse over the whole flight envelope is limited.
From this I understand that the "active cooling" is not something that the Chinese appreciate, let alone to pursue. Neither does NASA.
In a way, TRRE and TBCC are different animals from SABRE. SABRE is similar to
The problem with getting above Mach 5-6 is that once you enter that regime in the atmosphere you have even less time to scoop the air and even higher temperatures. Known reusable airframe materials have issues at, say, operating even at Mach 8. Titanium for example will lose its elasticity and stiffness and will deform at those temperatures. Let alone more than that. The whole idea about using a cryogenic fuel (it does not need to be liquid hydrogen, you can use liquid natural gas as well) is that you can use it to actively cool, well, everything from the engine to the surface of the aircraft. The SR-71 also cooled the airframe with the fuel for example. It is just that because the speeds were lower on the SR-71 the fuel didn't need to be at such low temperatures so Kerosene worked fine. Because you have to scoop more air at higher speeds, you start to see people switch to things like Waverider airframe designs, which are poorly tested designs. Plus the engine gets complex and heavy like heck, more than a combined turbojet+rocket like SABRE.
On the other side of things there are also designs like air-augmented rockets. The Soviets at one point had a ballistic missile design called Gnom and the modern Meteor air-to-air missile uses a Throttleable Ducted Rocket engine. Those engines have more Isp than a regular rocket engine because they scoop the oxidizer from the air. If the speeds are low enough and you only need to fly the whole vehicle once, like for a missile, you can basically make the whole thing of heat resistant materials that you know can survive the trip even if the whole skin or engine gets cooked and partially burned by the time you hit the target.
Folks working on rocket engines on aircraft which operate at Mach 20, like orbital space launch vehicles, also use ablative surfaces on disposable rocket engine nozzles. But those are typically more expensive, heavier, and less efficient than cooling the nozzle with the propellant. Checkout the history the SpaceX Merlin engine for example. The Soviets invented in the 1960s a process where you simply use a circular saw to machine channels into the rocket nozzle and then you just weld and braze an outer jacket on the engine. The propellant flows through the coolant channels before either going back into the combustion chamber to aid Isp, or is vented out, to simplify design. This is basically called the Sänger-Bredt engine design. It is quite cheap and reliable.