Detecting scheduling conflicts when events have a range of days is highly dependent on your base setup and your workflows. There are also tradeoffs between simply easing manual detection of conflicts (lower code) versus a completely automated system (more code).
You can combine your method of concatenating the name with the month with ScottWorld’s method of grouping by that field and looking at the summary bar for groups with multiple entries. However, this method does not scale well, and is problematic for ranges that span multiple months.
If you have a pro subscription and few enough items, you could try using a calendar view that shows ranges.
Unfortunately, Airtable does not have a built-in method for detecting scheduling conflicts.