For one thing, while there are HDMI-over-IP systems, it sounds like you are addressing HDMI over UTP cable which uses CAT cable but is simply a cabling interface and is not Ethernet.
HDMI also introduces EDID and HDCP issues that can require more advanced management than a simple Distribution Amplifier or DA. In simple terms, there is two-way communication between the source and display device and when you start looking at multiple sources and/or multiple display devices then supporting the necessary communication becomes more complex. For example, when a computer signal is routed to multiple displays how is EDID, in which a display communicates with a source to automatically select the appropriate settings, handled? There are multiple ways to address that but it is more than simply splitting the video signal.
HDCP is an additional issue especially if you are using HDMI and have any protected/encrypted content. That can also affect being able to convert any protected HD content to an analog output like VGA.
The underlying point is that what you seem to want is possibly not as simple as you make it out to be.