A lot of people Claim preamp has more influence on sound than amp.
Yes and these people are what we call, "wrong". No one component is any more or less important than any other. This extends beyond complete discrete components all the way down to component parts like caps, resistors, diodes, wire, etc. Even the chassis matters. Everything matters. Please see the System Description.
https://systems.audiogon.com/systems/8367
But in reality there are close to 10 :1 discussions on amps than preamp. What’s your thoughts?
Yes but that only highlights the inadequate education of the typical audiophile. One who knows the subject knows to compare everything based on sound, and knows everything matters equally, and there would be no such imbalance. Beware the audiophile bearing tech talk.
And how would you distribute your money between them? How much sound quality do you sacrifice for a same power integrated amp?
None. In fact it is the other way around. Its impossible for separates to match the performance of a really good integrated amp without spending a whole lot more money. This is because of the simple fact the two components require an extra power cord and interconnect. Only by not counting those in the total is it possible to come anywhere close in performance.
Those who tout separates therefore must always fall back on imagined gains in flexibility or reliability, or looks, or... well anything other than sound quality. Which simply cannot be matched without spending a whole lot more money.
Since we all regardless of budget are limited in some way it then becomes impossible to beat an integrated with separates. The question of how much you give up in sound quality is difficult to measure. But its there. We tried. We know.
As for the larger question of how much to allocate, that one is a lot harder to answer. Its not really about money. Its about sound. A really superb sounding tube integrated like a Raven Nighthawk for $4k could be matched beautifully with truly awesome Tekton Ulfberht speakers for almost three times as much, and connected with SR Atmosphere speaker cables that cost more than the amp. That would be divine because they are all exceptional and nicely matched even though nowhere near the same in price.
Likewise there are components like the Herron VTPH2A phono stage that are so good for such a reasonable price (under $4k) that it makes it worth the stretch if you are anywhere near there, and almost silly to go more unless you are silly loaded. The best systems are comprised of components like this, where each one offers exceptional performance at its price level, whatever that may be.
Ultimately, go by sound, not price. Use price as only the crudest first step in the process.