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

No opinion can be ventured without an overall budget. Can you spend $10K?  Credible products are available in each area for $2K to $3K. If that’s a stretch are you handy?  DIY options are out there. Don’t neglect your room furnishings and speaker placement.

In my experience this is impossible to determine. I currently use an Audiolab 6000CDT with Denifrips Ares ii total recommended retail cost AUD$2300, with Audio Physic speakers and 2 subwoofers RRP cost AUD$9500 and Vincent pre / power amp at RRP AUD$4800. So that's roughly speakers 60%, amps 30% and CD / Dac 10%. The fact is I got the speaker / subs for $4800, the amps for AUD$2600 and the CD / Dac for AUD$2000 so actual was nearer 50%, 25% and 25%.  So $ spend is what I aimed for looking at the market and how they each performed and matched. So the ratio depends on the equipment and deals you can get. I have always found bargains in speakers and never paid more than 60% of RRP. In this case my electronics were a rare bargain both open box but clearly never used (both sealed inside boxes) whereas the CD I bought because its great value anyway as was the dac getting about 15% off each. My second system is Audio Physic speakers list AUD$3000, paid AUD$1000 boxed and new (4 years old left at back of warehouse by mistake when they moved store), Naim Unitilite  (pre / power / CD/ Streamer / DAC / tuner) ex demo list AUD$4500 paid AUD $2500. So that's originally a 60/40 split at $7000 becoming roughly a 75/25 split on paid price. So making up a fixed percentage is just not possible if you are searching the market as you may buy separates or combined components as applicable/ available.

Hello cclee2022.  If the power amp limits you, you will not hear the total advantage changing components might give you. So, some might say that if the power amp is not great, your system will not "sing."  Conversley, there is less difference among good power amps than there is between preamps and CD players. Power amps are more important than cabling. Pruifi's new amp modules, used by NAD and many others are superb.  I use two stereo pair daily. At least as good, and I think smoother on the top end, is the Starke Sound A4-320, I have three of them. They are simply stunning. Home trials are allowed. Check their website. Last year, they put them on a "Black Friday" special. If they do it again, I will order two more. At the regular price (under $1500 last time I checked), they are the best bargain since the Magnepan LRS. On sale they are a no-brainer. I used to build my own power amps, not any more! I use electronic crossovers from miniDSP and DBX. Get the passive crossovers out of the amp/speaker path. You will be suprised! Most power amps have the ability to control the motion of the actual driver diaphragms ("dampong factor"). Coils and capcitors interfere with that. Also, the speaker cables are much less fussy when connecting the amp directly to the voice coils in the drivers. Are you "handy" with tools and small parts? Try the kits from Akitika. Happy listening!

What influences the sound characer the most, in order of magnitude:

1. Speakers
2. ROOM ACOUSTICS (often forgotten)
3. Amplifier
4. Source (CD player, streamer, DAC)
5. Cables

I'd spend my money in that order.

SQ is not proportionatel to expenditure.  Allocating expenditure by percentage is not likely to give you the best bang for the buck.  Pick components whose sound you really like.  Be prepared to dive in deeper if required and spend less on other areas where you don't discover a stand-out for you.

That said, I mainly agree with Pani.  Transducers are the area that influences SQ the most, that is record players, digital players and speakers.  Money spent there will have the most effect.  Then look at amplifiers.  For some, room treatment is a must, but it is best not to listen in compromised rooms, some of which remain compromised however much is spent.  Wiring comes last.  Whatever Miller says, it cannot influence SQ to the extent active components do.

Foolishly, Miller advocates equal expenditure on room treatment and wires.  That means you only spend 16.67% on players and on speakers, when if you were more sensible it would be 25%,  That's a stonking 50% uplift.

If I had to decide in the restrictive manner proposed by OP (and did not have a compromised room) I would go:
30% speakers, 25% player, 35% all amplification (less if you go integrated), 5% room, 5% wires.  And that's generous on the room and wires.

If you are digital believer and only run digital, I would drop expenditure sharply on the player (incl DAC).  True converts know that if you have sorted jitter, clock error and have a good DAC, bits is bits, so any decent player will do.  Read errors are almost non-existent on every player.