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

Everything matters. Amp matters a lot. Recently I’ve owned 3 amps I can swap among in my system. Each has a very distinct sound. All 3 are way more than enough power (120 - 275 Watts / ch) for my 96dB Tannoys, so overhead & drive capabilities are not really a variable. I’m even writing down particular combinations of cartridge & amp that work best together lol.

You didn’t specify your budget!  I can’t believe all these people made recommendations without that info.  It matters because up to a certain level, you’d be better off with an integrated vs. separate power and pre.  An integrated will compromise on the preamp section, but save $ on an interconnect.

I would think speakers would have the biggest impact.  I also think a pair of REL's make a huge difference, especially a lower listening volumes.

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!