Why doesn’t IMDB add plays to its listings?

Probably because this would be impossible to keep up to date and accurate. Plays run every day and multiple times on the weekends and the cast could vary on a per performance basis.

There are two websites for plays.

Internet Broadway Database
https://www.ibdb.com/

Internet Theater Database
http://www.theatredb.com/

IMDb also lists theatrical, commercial and other work in a person's (such as  Anna Kendrick)
Biography, Other Works and resumes.

Why doesn’t IMDB add plays to its listings?
Yes I know its a site made for movies
but there is no site that lists plays in any database format is there?
Movie Stars do plays too don’t they?
Posted 9 hours ago
by andrew
Joined on October 31, 2017
https://getsatisfaction.com/imdb/people/andrew_j5qvjzuuxu8k6
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https://getsatisfaction.com/imdb/topics/why-don-t-show-the-actors-actress-experience-in-roles-on-the...

Why don ́t show the actors/actress experience in roles on the Theaters
(not only movies and TV-series)???
Many actors/actress begin at experience in Theater...
why the IMDB don ́t show this, too???
Posted 2 days ago
by wolfgang amadeus rheinhardt
Joined on October 30, 2017
https://getsatisfaction.com/imdb/people/wolfgang_amadeus_rheinhardt

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Wiki Links....

add to reply by Dan Dassow, Champion

Other databases listed @
https://en.wikipedia.org/wiki/IMDb#See_also

Internet Broadway Database (IBDb) - https://en.wikipedia.org/wiki/Internet_Broadway_Database

Internet Off-Broadway Database (IOBDb) - https://en.wikipedia.org/wiki/Internet_Off-Broadway_Database

Internet Theatre Database (ITDb) - https://en.wikipedia.org/wiki/Internet_Theatre_Database
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https://en.wikipedia.org/wiki/Category:Broadway_theatre