A '78 280SEL?
Seems like an ignition problem that's causing a very weak spark.
1. Be sure the battery is fully charged, a (new) alternator doesn't
guaranty a fully charged battery.
2. Assure yourself that the engine's ignition system is working correctly:
That the distributor cap isn't cracked, that the rotor is OK and
that the coil isn't cracked.
That the ignition points (or whatever this engine uses to make and
time the spark) are OK.
Given your description, IMHO, you'll find the problem in the
engine's ignition system.
3. Don't tear into the ignition switch before you're satisfied that the
engine's ignition system is OK and that the neutral safety switch is
working as it should be.