It is trivial to make a deep throttling pressure fed engine for example. Like the Apollo Lunar Lander engine. A Flowmetrics engine is also trivial to deep throttle for much the same reason. It has no pumps just valves.
The deeply throttled engines you're talking about are not the same as the ones I'm referring to, which are those currently under development in China.
My definition of deep throttling is basically a 10:1 to 5:1 ratio, with 3:1 barely being considered entry-level.
I am specifically talking about achieving this on engines of over 100 tons of thrust class. The Apollo program proved that achieving 10:1 throttling on a 5-ton class engine was no problem. The Constellation program also worked hard to achieve 10:1 throttling on a 10-ton class hydrolox engine, but it never entered substantive development. As for engines in the 100N to 1000N thrust range, I recall the U.S. has achieved over 20:1 throttling capability.
The throttling I'm talking about must be achieved on a staged-combustion cycle engine, and it involves retrofitting the YF-100, an engine not originally designed for true deep throttling, to meet this standard.
Under the constraints of these three prerequisites, the technical difficulty of providing such an engine is far beyond your imagination.
Currently, the Raptor engine (over 100 tons thrust class) and a host of various advanced Russian engines on paper all claim a throttling capability down to 20-30% of maximum thrust (a 5:1 to ~3.3:1 ratio). But as far as I know, this capability has never actually been realized or put into practical application because there are many problems at both low and high power settings.
In reality, achieving 30% minimum thrust (a 3:1 ratio) in a real engineering application is already extremely difficult. Of course, if it were a gas-generator cycle rocket engine, the difficulty would be much lower. However, the specific impulse (Isp) loss would also be greater (this is the key to the Merlin 1D's success, and why China's equivalent-class engines use a similar design, abandoning the staged-combustion cycle).
In the deep throttling research for the YF-100, it's necessary to comprehensively adjust over 3,000 of the original engine's detailed engineering algorithms and formulas (which should be for the engine control system). China has done a lot of concrete work, conducting preliminary research on 10:1 throttling between 2010 and 2015 (with the earliest work starting in 2008). After 2015, it entered various stages of substantive technical verification, and literature from 2024 basically confirms that 3:1 throttling has entered full engine testing.
This is the real reason the CZ-8R hasn't materialized. For a 120-ton class engine, at least a 5:1 throttling capability is needed to achieve a single-engine landing for the CZ-8R. A 10:1 throttling capability is needed for a dual-engine landing.
Currently, the first stage of Starship essentially uses two engines for the final landing hover. This is because Starship's landing mass is over 240 tons. With two Raptor engines throttled to 50%, the thrust roughly balances its own weight, allowing it to approach the landing gear at a controlled speed of 1-3 m/s. If it used three engines for retropropulsion, the thrust (3x120 tons) would be 1.5 times the force of gravity, which actually increases the landing risk (making speed control more difficult).
The real reason the current Starship second stage recovery doesn't dare to land directly on the launch pad is imprecise speed control, not an inability to control the landing position. To put it simply, the current Starship second stage recovery has only achieved precise positional control on the X-Y axes. Precise speed control is needed on the Z-axis. The second stage also uses two engines, switching to a single one at the final moment, because the second stage's mass should be around 120 tons. If the Raptor on Starship could achieve 3:1 or 5:1 deep throttling, then 5:1 would be sufficient for a dual-engine second-stage landing, and 3:1 throttling would basically allow for a single-engine second-stage landing (enabling a controlled approach to a hover). With a 2:1 throttle, you would need to control the deceleration precisely (the thrust exceeds the vehicle's weight, especially since the second stage needs to reorient itself before landing, which is even more troublesome).
Therefore, looking at the current stage, the Raptor has not yet truly achieved 30% minimum thrust (3:1 throttling), but it has achieved 40-50% minimum thrust (a 2.5:1 to 2:1 throttling ratio).