Why isn’t connecting to your own local node a common feature for Bitcoin clients? I come from the Monero community and connecting to my own local node running on my home PC is very simple whether it’s on the GUI, CLI or a mobile wallet. 

Why does Bitcoin not have the ability to do this easily? Current using the electrum live server as a workaround but can’t help but feel this could be easier.
This is not an issue related to Bitcoin itself. Rather this is due to the protocols that clients use to communicate and the specific client implementations.

Electrum has defined their own protocol independent of commonly used node software. In order to serve Electrum clients, Electrum servers (nodes with software that can speak the Electrum protocol) have to also maintain more data than just nodes do which requires more computation and storage. Because this information is not readily available to all nodes, Electrum clients can only connect to these special Electrum servers.

You can operate an Electrum server yourself, it is just another piece of software to run. However only people who really know what they're doing and have the hardware necessary to run an Electrum server do so. By running your own Electrum server, you can configure your Electrum client to connect to your own Electrum server (and thus your own node).

The other commonly used protocol is BIP 37. Many clients that use the BIP 37 protocol actually do allow you to connect to your own node. However they typically also connect to many other nodes as well. BIP 37 has various issues and is being phased out in favor of BIP 157/158 Client Side Filtering.

Both BIP 37 and BIP 157/158 allow for clients to configure and connect to users' own nodes. It is just up to the specific clients themselves to allow users to configure this.