So let's suppose we charge everyone a connect fee for grid maintenance. That covers the cost of maintaining transmission systems, LV networks and excess unused capacity. It will also raise the cost of utilities for the poorest fraction of society. I was shocked to learn that there is a large segment of utility customers who use very little electricity. A connect fee would, for many of them, be a significant price increase.
Solar aside, people aren't really just paying for their usage. As you pointed out, part of the bill goes toward maintaining infrastructure. To the end user, this doesn't just mean that their usage is covered, it means that their usage pattern is covered as well. Some customers may not use much energy, but they may want to start a large load once in a while.
For instance, I don't use a lot of power, but suppose I came home and found three feet of water in my basement, so I go out and rent a couple of big submersible pumps. I run these for a couple hours, and my instantaneous power usage is around 4 kW during that time. Even if this is still a relatively small part of my bill (it would be about $1), it's still a lot more power than I normally use, and the grid has to be able to support that, and ideally it would be able to support that whenever I needed it.
Now, I think that the connection fee could be adjusted depending on the particular installation (eg, residential vs industrial, city vs country, etc), but I don't think it's necessarily unreasonable in and of itself.