If he is a 'ham' he is only allowed to broadcast on certain bands - or ranges of frequencies at restricted levels of transmitted power. Very few have such a low level of life as to be on all the time... :-)

Most hams are more interested in picking up low level signals from as far away as possible with minimal equipment.

"an increasingly complex aerial array"
Because of the range of frequencies, different aerials are needed for each band - and it is usual that only one band is being 'worked' (transmitted on) at a time. He may monitor many frequencies at once, but that involves no transmission of power.

"My wi-fi connection drops out more frequently than it used to"

It is not necessarily his fault. Many others things, such as OTHER wi-fiers may be moer likely interfering. While unlikely, 'false signals' MAY appear in that 'wi-fi' band - but I say unlikely, as such by-products reduce the power of the wanted transmitted signal on the desired frequency, something any ham would be eager to fix ASAP.

It is many years since I was up on the technical details - I never get around to sitting for my licence - I got busy with other things, so I cannot remember the full technical details now. From memory, the power would be less than the total of surrounding mobile towers.