If someone buys a good (and expensive) pair of headphones, why would they want to get a cheap amp? They might as well, just get cheaper headphones, and have more cash leftover, maybe to get a more synergistic amp.
I'm not saying that you HAVE to spend $1000+ bucks, but that's probably the easiest solution for someone who's already splashed $2000usd on their headphones, and apparently has little understanding of audio equipment.
There are different motivations in high end audio. Someone might get the most expensive stuff he can afford because he likes heavy nice looking gear or wants to impress his friends. But let's assume a different person who wants to get the best sound for a given dollar amount. For example for $2500 he can get an LCD2F with a $1500 amp, or an LCD3F with a $500 amp. Which will be better?
It is possible to build a solid state amp, using top quality components where it matters (in the signal path, I'm not talking about giant fancy volume knobs), of good design, with all forms of distortion below human audible thresholds, when driving any normal headphone, like an LCD3F, which is an easy load to drive. And produce it for well under $1000.
That doesn't mean it will sound the same as any other well designed and built amp. It does mean the differences will be subtle, and it also means whatever differences exist, are not necessarily "better" or "worse", just different. However, because of expectations in sighted listening, people tend to attach preferences to differences they hear aligned with price.
Consider for example the Meier Jazz vs the Bryston BHA-1. Differences will be subtle because both are very close to "the absolute sound". But let's assume a discerning listener could reliably tell them apart in a DBT. Knowing the BHA costs 4 times the Jazz ($1400 vs. $350), he would tend to assume the differences are in favor of the BHA. Yet without this knowledge - not knowing which was the Jazz or the BHA - preferences evaporate or split 50/50.
Key point: hearing a difference is different from having a preference. Just cuz they sound different doesn't mean you like one over the other. And just cuz you prefer one to the other doesn't mean you can tell them apart in a double blind test.
This is not mere supposition. Tests like this have been done many times and the results are consistent.
I don't mean to pick on these 2 particular amps - just examples meeting the requirement of "past a certain point". Both of these amps are well designed and built, using top quality components with measurable distortion below human audible limits when driving any normal headphone load like the LCD-3F.