You might consider replacing both the points and condenser!
Old points especially if they are re-gapped is asking for the symptoms you describe or worse and a intermittent condenser is also a pain. (cost of new is minimal vs the pain of having to go back and re-pull flywheel, etc)
When a condenser fails the unit will usually act like a bad coil and like Bruce inducated the points will start severly pitting, either on the ground or hot side. I've seen condensers fail by losing their capacity, (open) and I've seen them intermittently arc over (leakage) but it's not common, just a pain and hard to ID other than replacement on small engines.
I have Condenser testers, but I don't trust them on small engine type eq when symptoms are such as yours I sub the part. I mainly use the condenser (capacitor) testers when repairing electronic equipment.
Also replace the spark plug and gap it correctly. If that don't get it humming I've seen the Mag coils on H30's go intermittent. A new coil is about $25 from Napa and will require re-timing the engine correctly. The timing is usually very critical for correct engine performance. Also the H30 or H35 had two different cranks by year models and different timing degrees and if you retime with wrong timing specs per the manual, more pain. I think the crankshaft throw change was in the early 80's. I't's (the change) listed the the spec section of the Tech manual.
I would also suggest you drill and tap the flywheel for a 3 bolt puller (if it's not already done and the eq is your own, because go backs, pulling the flywheel several times can result is broken flywheel nuts, mangled crank threads or worse. The flywheel will drill and tap easily when off the engine.