If the recording is "flat" or at least as intended, then the playback should be "flat" as well at least to sound as intended at the volume level intended and you are going to get closer to that with a meter than with your ears.
You're confusing yourself. You're taking two things that seem on the surface very similar if not the same and thinking they are one and the same. They are not. Sometimes people think they understand, then do something that shows they really do not. This is one of those times.
One more time, with a lot more detail.
The word "flat" has two completely different meanings. "Flat" as measured with a meter is NOT the same as the "flat" that you call it when all the notes sound equally loud to you.
Sound is nothing more than waves of compression and rarefaction- high and low pressure. When that pressure wave moves a microphone membrane the mic puts out a signal. When the signal is equal strength regardless of frequency we say it measures flat.
But this measured flat is completely different than what we hear as flat. The Fletcher-Munson curves are really just a graphic representation of how we hear different frequencies at different volumes. Unlike the mic that "hears" them all the same regardless of volume WE DO NOT!
Just look at the difference!
https://en.wikipedia.org/wiki/Equal-loudness_contourNotice, we begin to hear midrange and treble frequencies at very low levels. Then as they get louder in volume we hear them getting louder in volume, and its almost linear. Except at the extremes. Low bass we don't hear AT ALL until it gets fairly loud. And then, see how the curves are all scrunched together? What this is saying, once we get to a certain volume threshold (which is quite loud!) then suddenly we become very sensitive to bass volume.
In other words, very small changes in bass volume matter a lot, but only once we reach a certain volume level. Below that volume level you can play with the bass a lot and hardly notice. This is why the old loudness controls were 20 dB. Think of it! 20 dB! That's huge! But at a very low volume its barely enough. Turned up real loud though, now as little as 1 or 2 dB is very noticeable.
So as you can see its hard enough (read: impossible) to ever get to flat, at least not without defining volume levels. (Equally true, you cannot get to flat without defining location.) These alone are hard enough to understand. But if you want to insist on throwing in "intended" as well, well then I am just gonna have to leave you to geoffkait. And good luck with that.