In theory, this sounds correct, but it is not. At least not from experience with many customers and my own WiFi networks.
First, the client needs a signal that is strong enough that it can 'commit' to. Boosting the power (legally!) helps here tremendously. A couple of dB's can make the difference between an iffy connection and a solid and reliable one.
Secondly, the router's antennae are (usually) much, much, much more sensitive than any normal client's antennae are.
This is why boosting the power on 'only' one end helps.
Yes, the signal may get distorted or noise is greatly increased when the Tx power is boosted. But that is a bad design/hardware issue. Not anything else.
For a client that is receiving a signal at the edge of where it can 'commit' to it, boosting the Tx level will help (even if the signal gets distorted or noise is increased too, up to a certain level).
How I see a router and client 'talking' to each other is more like this.
The router can talk louder and listen with more sensitive ears at the same time. The client just needs to hear and be heard too.
When we're talking about normal WiFi distances inside a home or office (usually just a few feet), the signal is still many orders of magnitude higher than anything that has been sent into orbit into our solar system and beyond. What the additional Tx power is doing is offsetting the physical/environmental obstacles and the self-imposed limitations (drivers, antennae design and cost in dollars and battery life) of our devices.
These devices are talking loud enough for the router to hear them (easily). The problem is that they really don't have the same hearing as the routers they're connecting to.
http://www.snbforums.com/threads/asuswrt-merlin-378-55-3_hgg-final-mod.26524/page-2#post-199549
The above link shows this in action. If the router really needed the clients to 'shout louder' too, the above results would not have been seen, even remotely.