Sorry, thunderstorm kept me offline.
Well okay here's the results so far.
I tried Alcohol 52% but it told me it couldn't run on Win98se so I must've gotten a version that's too recent. CloneCD however worked for both the problem disks, producing four files--one seems to be the actual image and I dunno what the other three are (Though two of them--a ccd and a cue--are mountable with the older version of DaemonTools I'm using).
Now, one of the games (its title rhymes with Piston Block You) plays just fine from an image, but the other (whose title rhymes with Feed or Bleed Fly Flakes) seems to be one of those weird "smart" games where even if you have the image mounted (and in fact installed the game using said image), it'll still ask you to put the real CD in. The only other game I have that does this is Risk II, and both come from before copy protection was a widely known thing so I don't know what kind of black magic this game has working for it.
Nero is still a no-go with these two games, but about the issue of startups... well, usually when I install Win98 one of the things I do after installing drivers and stuff is I go and use a program like CCleaner (an older Win98-compatible version) or RegCleaner to disable or delete all the stuff I know is garbage. So I didn't think of that... however, last time I ran both programs RegCleaner saw something in my startup that CCleaner didn't, which makes me both suspicious but also worried that it might be something really important so no its not disabled. I don't have that comp with me or I'd tell you what it saw, but I'll report on it later.
It's too bad CloneCD is a program you have to pay for and the version I'm using was an older one from before its owners changed so I can't even register it.
UPDATE: First, fixed a typo.
Second, the service RegCleaner saw in startup that CCleaner didn't was hidserv, which according to a Google search is just a human interface devices controller, and is likely the thing powering my optical mouse (my ps/2 ports went bad so I bought one of those PCI cards that adds new ones before realizing they just read as USB ports, but for my purposes it works anyway). So still can't find any explanation about Nero's sudden unwillingness unless this card is somehow causing it.
Also this Win98 computer never goes online at all so it can't have contracted spyware or a virus.
DOUBLE EDIT: Also, the same two discs Nero has issues with, I also tried with Omi, an MS-DOS program included with the SHSUCD drivers which I've had a lot of use for over the years. This only runs in pure MS-DOS mode (it won't even work in Dosbox apparently--at least, it doesn't for me) so there's no chance of something in Windows being the issue.... and yet, it has exactly the same symptom with these two discs: will start out fine then just suddenly quit.
The one thing it MIGHT be is the disks themselves as when I sent them through Isobuster it kept claiming there were bad sectors, but I'm not sure if this is normal for Isobuster to claim (in general Isobuster makes no sense to me).