How could we have so many past lives without reaching some degree of wisdom or enlightenment? While I do have a personal and intimate belief in reincarnation, I really struggle with the Buddhist belief that we have innumerably many past lives.

A monk I listened to explained that all the beings we meet everyday were once our parents in a specific past life, as well as our enemies.
Honestly, i find this perspective quite hard to believe.
It seems to me that if had so many past lives, we should have reached wisdom and Buddhahood at some point already.

How (according to Buddhism) is it possible to have such an accumulation of experience, without reaching some degree of wisdom or enlightenment?
Infinities can be one-way. Consider the negative numbers.

What the Buddha said is that there is no "first point".


  SN15.19:1.3: No first point is found of sentient beings roaming and transmigrating, hindered by ignorance and fettered by craving.


And with the ending of ignorance, one does not return.


  AN2.36:4.7: When they pass away from there, they are non-returners, not coming back to this state of existence.


Non-returners (anāgāmi) and arahants do not return. They have left behind the negative infinity of suffering.