Good morning @LorenMueller
Great question! There should not be any issues encountered using a later ArcGIS Pro client (such as 2.7) to interact and work with an earlier version of the utility network (such as UN v3). To put it another way, while you can use a newer client and server version to take advantage of new functionality, there is no obligation to upgrade the utility network's version to be "current" with the client or server release. While some capabilities and functionality may only be available with later versions of the utility network (such as working with nonspatial objects), this does not prevent the client from working with the earlier version.
The compatibility chart you referenced is primarily concerned with the version of the utility network created through the combination of client and enterprise during creation or upgrade. The version of ArcGIS Pro and ArcGIS Enterprise determines the Utility Network Version value when creating or upgrading a utility network.
Earlier ArcGIS Pro releases cannot open utility network datasets that participate in newer functionality. For example, you'd encounter issues if you tried to use ArcGIS Pro 2.4 and ArcGIS Enterprise 10.6.1 for sharing a utility network that has a Utility Network Version value of 4.
Hope this helps to clarify.
- Jon