(Slightly late answer ... I wrote this much earlier today)
I find it somewhat surprising that seemingly educated people even entertain the idea that a router with a 'powerful WiFi' would be beneficial. Isn't it obvious to everybody that WiFi is a two way communication?
What would be the point of your laptop/tablet/whatever being able to pick up your high powered router at long distance, unless the tablet etc was similarly high powered, in order to transmit back again?
It isn't as simple as that, though there is a balance required between the transmission side and the reception side.
Power is one feature for the transmission side. Sensitivity, and the ability to identify signal from noise, is a feature of the reception side. My direct experience comes from the GSM world, and I can say that there was always work going on to improve the reception side too. Frequency hopping and receiver diversity were such tools - primitive by today's standards.
In fact, high transmit power can be your enemy - particularly in a world where you have to share with other transmitters. Keeping the power down, but improving the performance of your receiver, is a *much* better solution. In WiFi, your neighbours will thank you if you turn your transmission down to be just enough to cover your home, but not theirs; you gain if they respond by turning their ower down. Win-win.
I imagine that the better DSP-driven NxN MIMO setups that we see nowadays are one response to improving receiver sensitivity in WiFi. 5G seems to get labelled with "massive MIMO", so we can expect better in future.
Beamforming - the act of using multiple antenna to pick out a target spatially - works in the transmit directions by ensuring in-phase transmissions arrive at the target. For the opposite direction, using powerful DSPs allows signals from multiple antenna to combine the "in-phase peaks" intelligently.
Looking at some data tables, it appears that better MIMO setups achieve better range and speed because of the improvements on the reception side, and keep those improvements at lower transmit powers.
(Adding in response to later posts: The MIMO improvement isn't really from improving gain via directionality; it is really from the improvement that you get in the smartness of the DSPs that drive the different antenna).
So ... where does that take us with advertising?
Unfortunately, the target for advertisers won't understand anything about sensitivity and spatial improvements from multiple antenna, and DSPs, but they do understand the concept behind "more powerful". For best WiFi performance, they probably need to buy the hardware that comes equipped with the best NxN MIMO antenna setup attached to the best DSP hardware, running the best algorithm. But the marketers can only really sell "more powerful".
But, as technical people, we should't hear the words "more powerful" and read it as meaning just higher transmit power. Maybe we just understand it as "more powerful DSPs".
Marketing is the ultimate "dumbing down".
There's a Cisco presentation on their WiFi available here:
Understanding RF Fundamentals and the Radio Design of Wireless NetworksIt is interesting in general, but the features appropriate here are in pages 45-55.