I'm surprised that no one mentioned the only-partial match "The Men Who Murdered Mohammed" by aLFRED Bester.
"Time Exposures" is a closer match to the question, but isn't the sought-after story.
So. Bester's story -- after years, I just re-read it to compare -- is less of a match; however it matches in three respects.
(1) Although not at the level of "humanity" finding time travel -- A professor and inventor "has found a way to travel backwards in time but not physically, just as onlookers."
(2) It is not quite that he went "backwards in time but not physically, just as onlookers."
He did, however, (A) fail to affect the desired result of killing his wife via timeline intervention, and (B) became more physically insubstantial himself every time he returned from a time-trip. This becomes evident physically, and with both humans on the telephone and computer voice interactions can not hear him.
"She will have ceased to exist,” Hassel muttered, blowing smoke out of the revolver. “I’ll be a bachelor. I may even be married to somebody else... Good God! Who?”
Hassel waited impatiently for the automatic recall of the time machine to snatch him back to his own laboratory. He rushed into his living room. There was his redheaded wife, still in the arms of a man.
And
He fought with the telephone, which seemed to weigh a hundred tons, and at last managed to get through to the library.
“Hello, Library? This is Henry.”
“Who?”
“Henry Hassel.”
“Speak up, please.”
“HENRY HASSEL!”
(3) "possibly in an older anthology: it was likely written in the 50s or 60s."
It was written in 1958 and appeared in several anthologies in the next few years.
There is, however, no mention of cameras or solving criminal cases. But rather, a funny story about a professor who became a criminal.
Or would have, except that his murders had no affect on anyone else's timeline.