As always, the "best" solution is the (1) one that works and (2) the one you can afford. I have several solid state switches designed to switch mains-voltage resistive loads such as incandescent lamps. I assume these are TRIAC-based devices that commutate (switch off) each time the AC line current goes through zero, as it does twice each AC line cycle. I've never tried to use these on DC loads because, unless the current is interrupted my some other means while they are conducting, they will not turn off after being commanded to turn on. Your mileage or kilometers may differ.
Or maybe your push-button switch needs to be a DPST or DPDT (more common) switch. You can use the other set of switch contacts to short to ground your System B signal, or raise it to 3.3V potential above ground... that is, assuming the switch actuation actually causes the 5V from System A to occur. In other words you don't need to sense that 5V is present from System A, as long as when the button is pushed System A is already supplying 5V. No relay, SSR or otherwise, needed.
In electronics, as is true for most problems, there is always more than one way to solve a problem, if there exists any solution at all. Some problems cannot be solved, but yours is not one of them. This forum is for electronics experimenters. We love to assemble real parts to perform real functions, and most of us know how to make use of what we happen to have on hand. Of course with China supplying the parts, and if you can tolerate the shipping delay as well as the uncertainty that the parts you buy from China will actually work, you can try many different solutions really cheap. But caveat emptor; you usually get what you pay for, so trust but verify. That may be why some of us (even me, sometimes) will simulate a circuit before actually building it with real parts. Nothing wrong with that if your intention is to actually build something, or to try to understand something with or without actually constructing it. There are no hard and fast rules in the electronics hobby except this one: stay away from the high voltage.