I think the idea is to keep some power going through the system so the CMOS don't wipe, but I certainly wouldn't do that as the second you disconnect the old battery you will get a spark/surge and then when you cut the new batter in you will get another spark/surge which could, and probably will, cause all sorts of damage.
BNT. Button cells have been used for longer, but the big issue is that some cells CANNOT be changed easily due to them being soldered down. Also, the heat from the "wrong" soldering iron could fry other parts. It's not always as simple as changing the battery on a watch, for example. But the cells are standard ones, and most boards do have them set in a holder so they can be changed. Buy a brand name system (Just think Dell, HP, F-S, etc), however, and then that situation suddenly changes (24 hours after the warranty runs out, iPods, anyone?).
Since the Ejector knows a techie, I say get him to look at it since he suggested it himself. Changing the battery is always cheaper than a new motherboard so if you CAN it's always the first option. But it could also be a sign of other problems, which is why, every time I clean out my system, I look at everything extremely closely via magifying glass and flashlight and check to see if any capacitors are getting fatter or there is any sign of something getting too hot. If I see anything like that, then it's backup QUICKLY time as something is about to go POP.
If Ejector can tell me what motherboard it is (Is it custom built PC or one from a major brand (need model number then)? The initial BIOS screen may say what the motherboard is, Gigabyte XY123Z, for example, before the ram test, but that depends on the BIOS so you may not be able to see that. There should be tools out in webland which will tell you, but my mind just went completely blank when I started to think of one. SiSoft Sandra, I think, auto detects the motherboard and it's either a free trial or a free program. Tells you all sorts of things about your system and may spot something.), I MIGHT be able to find out if the battery is replaceable or not.