He had forgotten to plug the television into the mains socket

Can't talk about American English, but in the UK the electricity that is provided via the national grid is referred to as the mains.

All electrical sockets on the national grid connect to the mains.

There is no main socket with subsidiary minor sockets, they're all mains sockets.

As far as your example goes, we're more likely to say, He had forgotten to plug the television in.

From Collins online.

Definition

mains electricity :electricity supplied to a building through wires.

Sample sentences.

The smoke detector should be connected to the mains electricity and fitted with back-up batteries which will operate if the electricity supply should fail.

The restored 17th-century farmhouse still has no mains electricity.

They are free to all those who decide to welcome the miracle of mains electricity into their homes.
Philip Marsden THE MAIN CAGES (2002)

There were several sheds, all in good trim, and two of them served by mains electricity, borne from the house on heavy duty cable.
Hilton, John Buxton PASSION IN THE PEAK (2002)

https://www.collinsdictionary.com/dictionary/english/mains-electricity

I wonder if (in fact, strongly suspect that) the electricity "mains" term came about because of the earlier water and gas pipes which were (and are) called a "main" - a water main is a pipe from the water company which supplies water to houses, and a gas main likewise supplies gas. They are big and buried under the road or pavement/sidewalk. So are electric supply cables in UK towns and cities. The electricity mains would be plural because there is more than one wire.

In fact a "main" is a big pipe supplying stuff or taking it away: you can have a steam main, a water main and a gas main supplying those commodities, and a sewage main to carry sewage away.

He had forgotten to plug in the 📺.....