I recently bought an old Ariens riding mower (pull-start), and after
cleaning out the carb, I managed to get it started.

However, whenever I take the mower out of neutral (not necessarily
into another gear, just out of neutral enough to disconnect the
neutral safety switch), it dies. I thought the purpose of this switch
was only as a safety for starting - shouldn't the magneto coils
maintain the spark after the initial start? To clarify: it is not
going dead because of any load on the engine - just the disconnection
of the safety switch.

In the meantime, I bypassed this switch (since its just wire in, wire
out, with a contact between the two), and now it won't start at all.
I feel like there might be something else going on besides just this
switch.

Any help would be appreciated!