I'd go for Socket 370, but honestly your workload could possibly even be resigned to Slot 1/A. If you want to go Socket 370, I would actually go early Socket A instead, as it's a FAR superior platform. Intel basically stuck beans up their nose and pants on their head for almost everything Socket 370 until the Core 2 series. It's the time when AMD really took off like a jet, and held the price to performance sweet spot until Intel eventually dethroned them with Sandy Bridge, and AMD took the Intel approach of sticking beans up their nose and pants on their head until Ryzen.
tl;dr 370 not good, 478 not good, do slot 1/a or socket a
In terms of period price/performance I fully agree (and was running the K7S5A at the time to prove it 😉 ), but looking back, and certainly with the games mentioned, even a late So370 system is total overkill. At this time Weebob already has both Intel-based systems, so price is zero. Getting a SoA system together wouldn't be expensive - they're 'old crap', not 'vintage' yet the way So7 and older have become - but would cost something, and it wouldn't actually improve performance in any sensible way with the stuff he's running. Plus SoA ran significantly hotter than So370, so more power consumption, more need for cooling, so more noise and more opportunity for boards or PSUs to decide to die (even if less so than with So478).
If the goal was an XP system, the RAM limitations of the i815 platform would be killing, but as a stable, reliable WIn98SE base I'd do So370 any day, sooner than an early Slot A Athlon or SoA Duron.