Yes,
I know black magic makes a $65 SDI in to HDMI box. Yes you can buy SDI switchers and most pro cams will have a SDI out but many average Church grade cams do not or may have both SDI and HDMI However, I have seen it get complicated as I remember my buddy doing an install and the Black Magic SDI box feeding I think a Roland switcher could not get the frame rates to be compatible but I can't remember exactly what the issue was, Aso with SDI cable you have different grades from SD-SDI, HD-SDI, 3GSDI , 6G, 12G
With HDMI I have seen something that should be so simple cause so many problems.
IMO, HDMI is fantastic going from say a home Blu-ray player to the one big TV with a 5' run and perhaps to a stereo or sound bar but when you start asking HDMI to do much than that it may work 100% or may go out or in and out for no reason.
It's my opinion HDMI was invented for one purpose. To replace the old red green and blue RCA video cables ( those cables had the same great picture ) and the red and white RCA audio cables and include all in one CONSUMER GRADE cable that the units you plug onto each end talk to each other to prevent illegal copying of Hollywood movies.
Then the Churches started using HDMI projectors and next thing you know entire streaming video systems were designed with a mix of CONSUMER GRADE video items but IMO HDMI can be problematic, especially Ama*** grade HDMI splitters. Baluns and such, I have seen many low cost China grade HDMI boxes last one year and just stop passing video.
However, I don't live in a perfect world so sadly lots of HDMI's are in it and I do run across situations like you have had.
I look at it this way... You wont find much if anything HDMI at your local PROFESSIONAL TV studio except the TV in the break room may have an unused HDMI port.
However when I so work with HDMI I have found in general most HDMI issues can be resolved by simply using the more $$$ HDMI devices from a pro AV supply. One thing I look for and have noticed, I have better luck with a HDMI box, splitter, converter, etc thats metal and not plastic. IF metal, it MAY last more than a year.
If your Church is now also using 3 other cams via HDMI and a Switcher using HDMI, you may just do best to roll with what you got and when you upgrade the entire system perhaps consider either a NDI networked video streaming system like I now use or design and install a known compatible turn-key streaming system via SDI,