If there is excessive resistance in the path of the power being provided to the starter, it will engage but turn over slowly. But this will not rapidly drain the battery as you describe. (It will drain the battery, but more slowly than you seem to be describing, as I read between the lines.)

If the engine is very hard to turn, the starter won't be able to do the job, and will drain the battery pretty quickly. A starter motor that is energized, but is being held from moving operates electrically essentially as a piece of wire. But you can push start it, so that doesn't sound like what's going on.

It sounds to me like the starter motor itself is defective. Remove the starter motor, and have it tested.

It also is surprising that your battery is losing power so rapidly. Is it really a new battery, or just new to the bike, but having been used? It seems you ought to get at least 15 to 30 seconds of battery power even in a situation such as you describe. That makes me think: Perhaps your bike is not properly charging the battery.

Measure the voltage on the battery with the engine running at least 12 to 15 hundred RPM (motorcycles, having high-revving engines, usually don't generate full voltage anywhere near idle speed). It should be at the very, very least 14 volts, and really should be between 14.3V & 14.5V. If not, it will have the power to run the lights and so on while the bike is not running--for a while anyway--but will not have the oomph to turn the engine over, and will certainly not have any reserve.

If the voltage is low you could have a faulty voltage regulator, or alternator.