Here are 3 different ideas:
1. Right click on node in properties pane to center
Yes you can right click on a node and center it but you have to go through the project explorer. When you have multiple runs and nodes, it is a bear to find in the project explorer. One of my suggestions was the ability to right click on the node name in the properties pane and be able to center to the node. This would save the user from having to search through the project explorer and lose their spot (I did this the other day and it is very cumbersome).

2. Click click in project explorer for center of pipe
When I said center of run, I misspoke. I should have said center of pipe. Nomenclature difference. We call it a pipe run. I will have to try and catch that.
I agree with you that a pipe such as a water line would screw this up. A simple node to node pipe is easy because it is azimuth from invert out to invert in and half the length of the pipe and add to the the invert out. Maybe you could still do this. It would satisfy a majority of pipe and even for pipes (not runs) that are not a simple point to point shape it would get you closer to the centroid of the pipe vs not have a quick way currently. The more detailed and the more nodes on a pipe, the closer you will get to the center of the pipe. It does not have to be exact, it is more to bring the pipe closer to focus especially with potential projects that will have hundreds of pipes on a project. For utilities project, this will be a huge asset.
I truthful see using the project explorer to QA/QC drainage and sewer pipes more than pipes such as water and electric. This is because we are given a sewer schedule and a drainage schedule that I am usually cross checking with the engineer. The names of these pipes networks are more crucial than with pressurized and cabled networks.
The picture below shows an idea of what I am purposing for a waterline line. I understand that the more the pipe starts to deviate from a linear shape, the further away the centroid of the nodes will vary. This will still get you close though. And typically (not as always) a pipe is thought of as a semi-linear shape because you need to apply a node (such as a bend fitting) when you have a hard transition such as a 90 bend.

3. Object Zoom Extents
Another thought is a sort of object zoom extents or fit to screen on a selected entity. This would fit that pipe to the screen and roughly put it in the center of the screen. Even with a center option you may only see a portion of the pipe. A zoom extents would tighten/expand the screen to encompass the whole pipe regardless of shape.