Why do you say that you pay more for the variable part?

On the UK out bills are two parts. The standing charge, which is the infrastructure rental, and costs the same wherever you draw 100A or 100mA, and the per unit charge. The later makes up the bill of your bill.

If I'm not mistaken, the UK uses price-cap regulation in electricity distribution, which is one where government just sets a maximum per unit price for electricity. The standing charge only pays the cost of maintaining his own individual connection to the network, but the fixed cost of installing and maintaining the distribution network go into everything else in the bill. So an UK's consumer "variable" electricity costs are actually the firm's fixed costs.

At various points both of the large parties have proposed capping energy bills. Not happened yet (the Tories ridiculed labour for proposing it, then a year later proposed it themselves)

We do have a very mobile energy market, switching provider is easy.

There may be price caps on the monopoly parting the system - the connection to the grid - that's because it's a natural monopoly, moreso even that the last mile of fibre from the cabinet to your house.

When politicians mention "capping energy bills", that means trying to cap final consumer tariffs. What I mean by "price-cap" is a regulatory model which determines how governmental contracts for public utility might be written. It's also called the "RPI-X" regulation model.

It does not determine final consumer tariffs directly, but instead establishes a maximum revenue per KWh for the regulated firms. And this model has been in place in the UK's transmission and distribution sectors since 1990.

Also, you cannot choose who distributes energy to you. If you could, the sector wouldn't even need regulation. But you can choose who supplies that energy. These are two different sectors, and they're regulated in different ways. The distribution sector is a natural monopoly, while the energy generation sector isn't.