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The desktop version of My Maps does have a search box and you can always toss a directions link in the marker description with the rest of the information. Quick and rough example here: https://drive.google.com/open?id=1XGn1LcjHiTkFQL7TQvRkBVvUxUM&usp=sharing
Though for 200 markers the smarter way would be to store the lat/lon coordinates for each place in a Google Sheet that you would then import into My Maps. You could even automate the creation of the directions URL by concatenating strings and using the lat/lon as the destination point.
HOWEVER this is probably all moot if you're wanting to view and search the map on mobile because in that case you're correct, there is no search box. For whatever asinine reason, there's just no way to search within a My Map on mobile. Not from Maps or even the My Maps app.
So your options now become a.) learn to code and make your own app with the Google Maps API or b.) find some other mapping application that has this specific functionality. I haven't tried them, but I found a couple of apps that may fit the bill: Save Location, My Location GPS Maps