It's very possible. Most pre-Disco bikes were 100% mechanical plus bare-minimum electrical for street-legal reasons like brake lights. Electrical accessories like a starter came way before EFI systems.
mechanical gauges, mechanical fuel pump, carburetor, mechanical throttle cable
Check, check and check. My 2006 bike worked like that. My 2013 bike still has a mechanical throttle cable. Traditional-style bikes (and some scooters) with the fuel tank in the "normal" place and a carb don't have fuel pumps. Gravity handles fuel delivery to the carb.
push starting only
You have apparently forgotten (or are too young to know about) kick starters.
Regarding ignition, you do not need an electrical system for spark plugs. You need a magneto. Self-contained gizmo that generates the necessary energy for the spark. Yes, it's "electrical" but it's very simple and single purpose - it doesn't run on 12V. Modern light aircraft still use magnetos for ignition as they are extremely reliable - all they need is engine rotation. This minor exception eliminates the considerable difficulty of a small-displacement diesel engine.
It seems what you want is a not-too-new Harley minus the headlight. Drive in daytime, stick your arm out to signal turns and it might even be licensable.