For efficient speakers, how much difference does an amp really make?

Current is king. An amp that is comfortable at 4 ohms and stable to 2 ohms will be able to drive those with authority. An amp that is stable at 4 ohms and nothing below will likely sound thin as volume increase. I had this problem in the past with an integrated that was 4 ohm stable but not really happy there and with a power amp.  
At the time I upgrade to a Rogue Hydra (100w class D and doubles output down to 2 ohms) and my speakers came to life.

Because I import Art Audio, I brought in a pair of $17K mono-blocks and not shockingly, they have plenty of current to handle my lower impedance speaker.  

many manufacturers put in a power supply that delivers a big number into 8 ohms but won’t be up to the task as impedance drops. 
At times you can find that Hydra in the $1500 to $1800 range.  A stereo 100 which will have 50w in triode, 100w in ultra-Linear would be another great choice for hybrid and tube gear.  
For Solid State, Musical Fidelity is a strong value and does well at 4 ohms.  If you can find a used M6 PRX, they are amazing.  A new one is $2300 I think.  The older ones were $3K.

My take - the more efficient the speaker, the more the balance shifts toward quality from quantity when it comes to amplifiers. 

For a given budget, any amp designer is going to have trade-offs between power and finesse. There are obviously going to be some designers/companies that are more creative/efficient, etc. but building a more powerful amp means higher voltage parts, bigger transformers and caps, bigger heatsinks, larger chassis, etc. which take budget away from higher quality components. 

In other words, high efficiency speakers do not, in the least, diminish the benefits of a high quality amplifier, but they do reduce the need for tons of power. If anything, a high efficiency speaker will expose the limits (noise floor, linearity, etc.) of your electronics much more readily than a low efficiency speaker. 

If you've got high efficiency speakers, take advantage of your lower power demands to get an amplifier with more refinement for the same budget.

High efficiency speakers done well is a good thing but good quality full range ones are big bulky and thereby relatively expensive and generally require a lot of room to breathe.  Not for everyone. 
No single approach can do it all best.  Hence the variety. 
Also, these days, high efficiency Class D amps are a game changer delivering more often for less and in a smaller easy to handle package that is practical for many.  

Yes a good powered sub or two set up well is a very effective way to address power needs. Power needs for flat extended bass increases exponentially as  frequency decreases which is why most good subs these days deliver 100s of watts often using Class D amplification. Plus use of a crossover that relieves the bass load on the mains usually results in better performance from those as well because now your main amp has to not work as hard as before trying to do it all. Best thing since Swiss cheese......