Can VOR indication be reliable if one of its antennas breaks?
No, it should not be used.

Twin antennae, as you have shown, are balanced and fed into a balun (BALanced to UNbalanced) to feed the single transmission line to the receiver.   Together, they provide the correct impedance for the aerial system.  With an antenna broken, it is likely that the impedance of the circuit is wrong and therefore, the input to the receiver is, at best, suspect.

[GUESSWORK]

It's impossible to predict the indicated behaviour without knowing precise details of the aerial fit but a reasonable guess would be that the standing wave ratio in the feed will be altered and therefore, a phase error will be introduced.  The magnitude of the error will vary in proportion to the sine of the actual phase angle so my guess would be that the bearing will be accurate when due South or North of the VOR then increasingly inaccurate as you move round with an maximum error at 90 or 270.

[EDIT]

Early in the morning when I first wrote this but I think I am wrong about being accurate due North or due South.  It will be accurate somewhere on the compass and at 180 degrees opposite to that, with increasing inaccuracy as you move towards 90 degrees away from that somewhere.  Only calibration would tell you where the somewhere is.