do you have a straightforward advise for me that Me as a novice can understand and implement?
Excuse me for providing a to detailed explanation.
I try give a more end user friendly description of what happens.
A device you connect to the USB port of a computer may just show as a single device in your operating system (standard keyboard or mouse) but it may also have several virtual devices inside that all are operated of the same USB connection (keyboard with integrated trackpad, multi-functional printer that also shows up as a scanner, etc.).
The Leonardo always activates a virtual serial device when it's connected to a computer. During startup this is directly available for uploads for a few seconds, after that the programmed sketch is started. Once the sketch is started not only that sketch runs but the IDE automatically adds code to the uploaded program to have a virtual serial device running in the background, even if you do not use the Serial object to send or receive data from the controlling computer. That device is only there to listen for a connection. If that connection is at 1200 baud and the connection is immediately closed (without sending any data), that's the signal for the Leonardo that the IDE wants to upload a new sketch and it has to automatically reset. If that functionality is not included you could upload one sketch over the USB connection and never use it again for sketch uploading. This is not what most users would expect so the Arduino people implemented it the way that most users feel happy.
In your case I guess that the KVM switch has a very simple USB host interface. If a USB device is connected that does not only identify itself as a keyboard, it simply ignores it and doesn't react on any message received from that device. That way the KVM device doesn't have to implement the complete USB software stack but can keep that part small and rather simple (I guess a connected USB hub would also be ignored).
You can program the Leonardo to simply identify itself as a keyboard (and not serial stuff) but you cannot do that with the IDE (to my knowledge). And you cannot upload the so built program by USB but you have to use an ICSP programmer to directly write the processor's flash. Unfortunately this is out of a novice user's reach and I have no better solution for that.