I could not disagree more.
The transducer is the part of the chain that makes the greatest audible difference.
You can get musically satisfying sound from a Magni/Modi through a pair of LCD-X, but I don't care if you have an Esoteric DAC and a Studio6, you're still not going to save a pair of Beats.
The rule of thumb for decades in speaker-fi has been to begin with a total budget. Then allocate HALF to the speakers and the other half to source and amplification electronics.
The beauty of head-fi is that the best transducers are still not going to cost what the best speakers will, relative to the electronics. This means that you can allocate more money to the electronics, but you still need to start with the headphones.
This is NOT to say that you shouldn't spend as much as (or more than) the headphones on the amp. It's simply to point out that final link in this particular chain is the most important.
In this case, the LCD-X is showing itself to be a worthy anchor to a musically wonderful system. If you start there, even with inexpensive amplification, you have a great launching point to a system. You can make subtle improvements along the way with source and amp changes, but the transducers will always give the electronics the opportunity to showcase their worth...or lack thereof.
Big +1 to this. One of the best headphone amps I have ever heard at meets is the Bryston, for me it outperforms a number of very expensive headphone amps (including all tube models).
Historically the rule you state has been the recommendation in the US, while in Europe they tend to buy less expensive transducers and spend more on electronics. I am sure the trend has gotten much more blurred recently though with no consistency either in the US or in Europe.