Apparently the original game and Brood War expansion are free to install through the Battle.Net launcher these days.
If you have the original discs, the later official patches added the ability to copy the "mpq" files from the CD into the game's directory, so you no longer need the disc in the drive. Of course, you're still going to need a drive for the initial installation. That should work for single player (it's been a few years since I last did it) but I don't know about online multiplayer.
I don't think it's the chips, but the operating environments. Modern CPUs offer dozens of multipurpose registers and many more instructions and addressing modes compared to those old, low-cost CPUs, which should make things easier, not harder. But no-one's building old-style dedicated systems around modern CPUs; our code now has to play nice with firmware, OS, libraries, and other processes, including resource management and preempting.
Compare a single-gear go-kart to an automatic sedan. Getting top performance out of the go-kart on a closed track is difficult and requires nuance. If we could drive the automatic sedan around the same closed track, we could easily demolish the go-kart, and not just with raw engine power. The improved acceleration, braking assist, and power steering are enough. But when we drive the sedan we're usually doing it on public roads with traffic signals, intersections, speed limits, and other road users. That's what's more difficult.