Event Conflict Detector

You are provided with a list of events, where each event is represented as an object with a start time and an end time. The times are given as integers which represent minutes from the start of a day (i.e., from 0 to 1440). For example, an event may be represented as { "start": 60, "end": 120 } for an event starting at 1:00 AM and ending at 2:00 AM.

Your task is to write a function that takes the list of events as input and performs the following:

1. Determine if there is any conflict between events. A conflict is defined as any pair of events that overlap in time.
2. Calculate the maximum number of events that overlap at any single point in time.

The function should return a result that indicates whether a conflict exists and, if so, what the maximum number of overlapping events is.

Assume the list of events is unsorted and the size of the list can vary. Implement an efficient solution that handles both small and large inputs.

Please provide your solution along with any tests you would use to validate its correctness.