That is what the wirecast solution will do. To answer your question, yes, a computer that has multiple inputs will do what you want it to. Also a video switch/mixer will do that as well. Pull your feeds into the video switch and then use a single output to the VidiU.
In this way the VidiU and the Webcaster are performing the same functionality (only with a reliance on HDMI input instead of Composite-Video.
In our church, we have a tendency to go over-the-top at times and I think this is good council to follow. I think if it becomes too much of a burden putting on something fancy-dancy just 'cuz' then we go beyond what we really need to be doing.
In my former stake, we had a member that routinely did remote camera studio-type work, he had a couple of quickly portable nice HDMI cameras. I had 2 different members that did camera operator work in College (one with BYUTV) and to them it was old hat, in fact, I had to hold them back a bit, but providing text overlays, hyms, etc... multiple camera angles, all that came very natural to them and didn't really constitute a major effort.
“A long habit of not thinking a thing wrong, gives it a superficial appearance of being right, and raises at first a formidable outcry in defense of custom.”
― Thomas Paine, Common Sense