The coax output is not feeding the projector, just the TVs. They are mostly old CRT sets. The problem I am trying to fix is that the VCR is malfunctioning and needs to be replaced with something that can do the switching duties. I suppose I could find another old VCR and call it good.
Ok, the VP-728 might be able to function as a switcher for your setup. The manual does refer to "RGB
HV/Comp/YPbPr Output Signal" specifications. This leads me to believe that the HD15 output port can be used with a 5 wire breakout cable and then configured to output a 480p signal via composite video. The next question would be if you are feeding the TVs a RF modulated signal. If you're connecting to the TVs via the antenna connector then that's a yes. In that case you'll need a RF modulator. Another clue is if you have to tune the TVs to a particular channel in order to view the content. My guess would be channel 2 or 3. Fortunately Blonder Tongue analog RF modulators (BAVM series) are super cheap on ebay these days, under $100 for sure, maybe even under $50. You'd need a composite video DA to split the signal between the modulator and the projector. Also super cheap on ebay for a quality rack mount unit.
Now if you go this route keep in mind that just because you'd have HDMI inputs doesn't mean they would work for you. You still have HDCP to contend with and any source that had HDCP encryption on it won't output through the analog output. Also, while the unit will scale high res down to whatever your output resolution is it may not be pretty.
This isn't particularly hard. It'a fairly simple signal flow. I'm an audio guy who's managed to master video world as well. It just took a lot of study and patience. If you want to get it figured out and if you're a competent audio plumber then this should be completely in your capability.