Ceiling – How to determine whether the existing fixture can support a ceiling fan

You have two issues:

Strength of the hickey - Some hickeys are cast from cheap pot metal and would be sketchy. Some are steel but bent in the shape of a U with one open side, also problematic. Heavier cast iron ones with support on both sides wold probably be better.

Strength of the crossbar - Also an issue. consider a heavier duty type or even a full round. Then consider doubling the cross bar up with small bolts joining the pair.

Supplement - In line with other answers and comments, there are real risks installing a fan with other than a properly rated fan box. Perhaps most important is the security of the box itself. The fan depends on the plate, the plate on the hickey, the hickey on the stud and the stud on the box. Is the box well enough attached that it can't be loosened by fan vibration? Also, if the plate is not pulled tight against the box, the fan is balanced on the single point of the extension coming from the hickey.

The threaded rod you see in the middle is a stud. It is used to attach mounting bars or hickey (a connecting device). Studs are attached to the box, not to framing members. It is no indicator of the strength of the box itself, or the attachment mechanism.

Fan boxes are characterized by their screw system that ensures a redundant attachment between the fan and the box. They also must be solidly mounted to framing members.

There are no heavy screws showing in the interior of the box, and the screw in the clamp holding the NM wiring going straight up suggests that there is not a framing member directly above the box.

It is both against code, and more importantly, dangerous to mount a fan in an unapproved box. The weight an dynamic load of a fan makes failure a high risk.

You need to replace the box with an approved box and ensure that the box is firmly attached to framing members. It may require some drywall surgery, but it is worth it.