I think this happens more times than we might think. I have had three pusher caps fall off various chrono watches in the past 3 years. Usually older style watches that are more likely to unwind the pusher from its cap. Most modern watches capture the pusher cap so it can’t rotate. It’s the rotation that tends to encourage unwinding of the pusher stem.
If it was stiff when trying to operate, then I would definitely send it to a repairer in your neck of the woods for assessment. The chrono lever arms can sometimes malfunction and that might make it stiff to use. Lucky you were able to retrieve the pusher cap and not lose it. Hope you did not lose the return spring either. If you ever have the opportunity to take a watch apart, you will usually find some form of screw end in the pusher stem that allows you to screw in tight, the stem onto the winder, but from the inside of the case barrel. Whoever designed this arrangement originally………………….
Probably the same person that designed spring bars..........