Would you want to be the person on the other end of this phone call? (Smart Broad Films/Sundance Selects)

“Elaine Stritch: Shoot Me” could easily have been nothing but fun. The documentary’s subject has been a theater, film and TV legend for close to 60 years and she’s an absolute riot (non-Sondheim geeks might know her best as Jack’s mom from “30 Rock”).

Stritch storms around New York, often with no pants on, and still performs a nightclub act across the country, often with no pants on. She is funny, she is sharp and it would be a joy to just watch her do her thing for 90 minutes.

And while “Shoot Me” director Chiemi Karasawa shows us all the joy that Stritch undoubtedly feels, the film is at its best when the darkness seeps through.

Some of the sadder moments come from external forces — Stritch is hospitalized multiple times in the film, usually for complications from diabetes — but the moments Karasawa handles most adeptly are those that come from Stritch herself.

Stritch says in the film she’s an alcoholic who was sober for more than 20 years — sometimes she says it’s 22, sometimes it’s 24 — but now she allows herself “one drink a day” (she has a fondness for cosmos, served in lowball glasses that make them look more like juice than cocktails). Which, fine, when you’re nearly 90, I think you should be allowed to pretty much do what you want.

We see Stritch on her way to an AA meeting, we meet a woman who talks about how important Stritch was to her own sobriety, and we see Stritch show off the mini of gin she always carries in her purse. She talks constantly about her diabetes (on the set of “30 Rock” she greets fellow diabetic Tracy Morgan with “How’s your blood sugar?”) but her disease seems poorly managed, and the booze probably doesn’t help. She slurs her speech sometimes; is that the alcohol, the diabetes or age?

“Shoot Me” deals with Stritch’s drinking in intriguing ways. First, it’s not the focus of the film; few movies give addiction second or third billing, as it gets here. Second, there’s no judgment. Finally, the film doesn’t treat alcoholism as a black-and-white issue, whereas most alcoholics in movies are either stone-cold sober (and kind of prissy about it) or pounding fifths of vodka as fast as they can.

I don’t know if Stritch’s drinking is a problem; alcoholism is such a strange, nuanced product of genetics and psychology and behavior that I wouldn’t even venture a guess. And Karasawa never presses Stritch about her drinking. Not that it would matter — as the old joke goes, “How do you know when an alcoholic is lying? Her lips move.”

When it comes to drinking, Karasawa lets Stritch’s lips move and move and move and, in doing so, gives us the most honest moments of the film.