Can a small household popcorn maker heat the syrup directly?

Absolutely not! Directly pouring syrup into a small household popcorn machine for heating is a very dangerous and damaging operation. Don't try to save time, explain the reasons in detail:

Syrup turns into 'glue', the machine strikes directly:
When syrup gets hot, it immediately becomes sticky and thick, like strong glue. It will tightly wrap around the stirring blades (if any), making the blades unable to rotate or even stuck, and the motor working hard may burn out.
The corn kernels are covered in viscous syrup and cannot be heated to roll and expand, resulting in a large lump of uncooked, clumped sugar that cannot burst out into popcorn.

Instantly burnt and thick smoke billowing:
Syrup is extremely sensitive to temperature. The high temperature of the heating plate/pot of the rice blast machine far exceeds the range that the syrup can withstand, and it can burn and turn the syrup into black charcoal in just a few seconds.
Burnt sugar not only produces pungent and choking thick smoke (triggering a smoke alarm is not surprising), but also releases a bitter taste that cannot be eaten, destroying the entire pot of corn kernels.

Cleaning up becomes a 'nightmare':
The syrup that sticks to the walls and leaves of the pot at high temperatures becomes as hard as stones after cooling (think of the burnt sugar chunks stuck to the bottom of the pot). You have to use a knife to scrape or force to pry, which is highly likely to scratch the non stick coating or machine liner.
Does syrup seep into the gaps and air vents of the machine? That basically declares the machine to be scrapped and impossible to clean.

High risk of short circuit fire:
The boiling syrup during heating is prone to overflow and flow into the circuit board, motor, or socket interface at the bottom of the machine. Sugar is conductive and has a high probability of causing short circuits, sparking, or even catching fire.
Syrup drips onto the shell or tabletop, making it difficult to clean after cooling and hardening, and may also burn people.

The correct approach:
After popping, add sugar: First, fry the original flavor of the popcorn obediently. While it is still hot, pour the melted butter/syrup (melted separately in a pot or microwave) evenly over the popcorn, and then quickly stir and mix. Although it's a bit troublesome, it's safe and delicious.
Use pre mixed caramel corn kernels (caution): If you insist on caramel flavor, buy caramel flavored corn kernels specifically labeled for use with a hot air popcorn machine (pay attention to the instructions! Ordinary pre coated syrup is not suitable). This type of corn kernel has a special high-temperature resistant sugar powder on its surface, which is definitely not suitable for ordinary syrup.