How much does power amplifier really matter comparing to other hi-fi components?

Clearly, speakers are central. However, you can get and use a perfectly great speaker that is not that pricey. An “honest” transducer need not have the big scale of a large, expensive speaker. And matching your speaker with your amp is also central.
Percentages are not important or relevant. Having the appropriate speaker to amp match is.

I love the question.  What you didn't ask about was the way more expensive room treatments, cables, power conditioners, isolation pads, racks, and subwoofers.  Those actually cost more than the components in question in both my listening rooms.  

To answer your question, I have a dedicated listening room and nearfield setup in my office.  I'll start with the big system with BIG speakers that unsurprisingly ate up 40% of the total cost despite answer 1 equipment, the DAC/clock/upsampler stack being absolutely ridiculously expensive, possibly the most expensive, and only accounting for 9.8% of the total budget:
1) 9.8%
2) 19.1%
3) 31%
4) 40.1%

The little nearfield office setup does have small-ish speakers.  But they are possibly the best nearfield speakers and are shockingly expensive for their size, yet answer 4 this time is only 8.5% of the total cost:
1) 17.6%
2) 28%
3) 45.7%
4) 8.5%

Cheers!

Responses received are eye-opening. Learning the motto of “it all matters” and the various distribution of components / contributions for a true hi-fi experience can be a great option s reference for beginners like me.

It all matters from the perspective you need these components in order to create a system. Most of the answers you got are nothing but audiophile nonsense. The amp, preamp, DAC if you go digital are way less important than the speakers and room. If you want a true high fidelity experience just spring for a pair of  Dutch and Dutch 8c and a decent streamer, apply EQ accordingly and room treatments. If you want a passive speaker most of your budget should be speakers and room treatments. As long as you get speakers that are not crazy to drive there are literally dozens of amps and preamps, DACs that will work and don't cost an arm and leg. 

TAS Robert Harley’s book “The Complete Guide To High-End Audio v6” book has an budget example of 36% speakers, 50% amplification+DAC, 9% power conditioner, and 5% interconnects+ cables.  But these percentages are not set in stone- you may get a great deal or discount or may want to pay premium for desired component.  But maybe having a place to start vs no clue is a good thing.