Is it better to use a WiFi channel with high frequency?

Channel numbers do not denote power and so channel 11 is not "better" than channel 1 simply because it is 10 digits higher.

WiFi does have overlapping channels, which means that devices don't want to be on a channel that is too close to another nearby stations channel. For best results and interoperability (least interference) there are only 3 channel choices: channel 1, channel 6 and channel 11. Here is an image showing why:

If there are many networks nearby then you want to choose the channel which has the fewest or weakest signals in it. If, as you mention, that happens to be channel 1 then that is the channel you should use.

The answer is "sort of" and "it depends".

If you are talking about the 2.4 GHz band - and this would seem to fit your description best - then he is pretty much talking garbage. The best band to be on is the one with the least noise - both on the main frequency and surrounding frequencies. In this case, Channel 1 sounds better and the tech sounds like he does not know what he is talking about.

If you are talking about the 5 GHz band, its a different story. Different channels in the 5 GHz band have different channel widths and different maximum transmission power - thus in the case it may very well be true that (some) higher frequencies do actually have more power. That said, the channel numbers don't match up very well with what you have posted (Channel 1 does not exist, 7 and 11 are generally not legally useable and are low bandwidth anyway)

Given the same transmitter output power, in the absence of ANY interference, the lowest-frequency channel will nearly always be the best. This is because higher-frequency radio waves do not penetrate matter (including air) as well as lower-frequency waves.

It is actually probably true that channel 7 is the most powerful channel. The hardware amplifiers and filters in WiFi equipment probably use something around 2.447GHz as their center frequency. Depending on filter design (bandwidth, filter type etc) and other characteristics you could probably see quite a bit (3dB?) of power loss at the outer channels. Will you be able to notice this much roll off? Probably not for innumerable reasons like noise floor and reflections.

Technically amplifiers get slightly less efficient (powerful as they increase in frequency so 1 should be more powerful than 11 if the center frequency is exactly between them, but this is almost certainly in the noise.