So many babies were abandoned in Medieval Europe that many churches provided drop-off places for mothers to safely leave newborns so they wouldn't be taken as slaves or killed by the elements/wild animals. What happened to these kids in the church's care? Were they all destined to be monks?

Orphaned children left in ecclesiastical care throughout the Middle Ages are a very broad topic as the Middle Ages itself is a quite broad time period.

We can say for sure that some periods saw quite an increase of orphans being taken care of by churches and monasteries, especially in the case of Italy after the XII century roughly speaking. We are able to tell by the usage and spread of certain surnames which can be linked to the condition of having been raised an orphan by what we can call a welfare institution. Italian surnames Esposito, ("[having been] Exposed", left out in front of a church or abbey), Proietti ("Being projected [out], being left outside a church), Allevi/Diotallevi ("Raise/May God raise you") and similar Diotiguardi ("May God look upon/after you"). First names such as Fortunato ("Lucky") can also be found in records about orphans chiefly during the Late Middle Ages. The above mentioned surnames are just two out of many dozen examples which have regional variations according local dialects.

An orphan child would not necessarily end up being a monk however. Adoptions of orphans (what we can consider adoptions, in any case) was a widespread practice in the cities of Medieval Italy, in the sense of families taking into their care a child which was previously looked after by either a church institution or a lay one.
I would like to point out that many churches were in some form "attached" to hospitals, which were, in turn and in most cases at least in Italy, lay institutions. As an example, the hospital of the Annunziata of Capua (which was the second most important city in the kingdom of Naples during the Aragonese period) had at least eight priests among its salaried staff members according to a registry of its financial activities for the years 1477-1478. We also know that just beside the hospital's main building a church was built in the same moment the institution was founded.
Said registries keep track of the orphans the institution had taken in and was raising, mostly by recording expenses for food, clothing, footwear and the like. The aforementioned salaries of the employed personnel at times state that former orphans stayed at the institution to work as labourers. Orphans were adopted of course, mostly it seems by craftsmen or families of high status. In the first instance, it's not clear if orphan boys were previously employed as apprentices and then formalized their status by getting adopted by the craftsman, but a link between the fosterage and the apprenticeship has been confirmed. In the latter, it's often the case of orphan girls who will serve as maids, however with the promise by the "foster family" written on the contract to provide her with a dowry when she marries.

In short, not all orphans who were taken care of by churches or other religious institutions would become monks. My personal theory is that taking vows would have been a rarer eventuality than being inserted back into society in some way or the other.

Sources:
Marino, S., 2015 Late Medieval Hospitals in Southern Italy. Civic Patronage and Social identity, in Mediterranean Chronicle, vol. 5, Diavlos, pp. 141-159,;
Senatore, F. 2021, L’Annunziata di Capua alla fine del ’400: l’ospedale e la sua attività attraverso un registro contabile del 1477-1478, in Quaderni dell'Archivio Storico del Banco di Napoli, vol. 1, n. 3, pp. 81-120;
Marinò, M., Senatore, F., Tiseo, M. P. 2020, Quaderno dell'entrata e uscita dell'Annunziata di Capua (1477-1478), in Idem;