TL;DR: The phrase is common and familiar within the theatrical world. It dates from at least the 1880s in the US, and was in use in the UK soon after that.
- Apart from the movie cited above, is there more evidence of “angel”, with the above connotation, used in theatrical contexts?
Yes; the use of "angel" to mean an investor in the theatrical world is quite common and venerable. It shows up in lots of old movies set on Broadway, for example. One such example off the top of my head is 1933's 42nd Street, where the star's sugar daddy is financing the play; when he finds out she's two-timing him and wants to pull his support, the cast and crew try to sweet-talk:
You've sunk $ 70,000 already, and you're gonna toss that away...
...because of a dame?
. . .
I wish you wouldn't take it like this. Back in New York, they're calling you "The Angel of Broadway."
—42nd Street (1933) movie script via Springfield! Springfield!
And a quick Google or news search for "theatrical angel" or similar will turn up plenty of contemporary examples, as well. For instance (emphasis added):
The economy has dropped faster than a falling angel, and investors for live performances are growing ever more scarce. . . . So Ms. Clarke returned to the phone, desperately seeking a theatrical angel. . . .
Ms. Clarke said: “Rhoda Herrick [the investor they found] came like a genie in a bottle, after four years of bloodied knees and scratched elbows. In this economic disaster, that this woman had the interest and ability to give this piece a life — it’s kind of like a fairy tale. I pinched myself that I had somebody who believed in the project.”
Ms. Greenwood piped up, “That’s why they call them angels.”
— Patricia Cohen, "How an Angel Came to Cultivate Martha Clarke’s ‘Garden’", The New York Times, Nov. 29, 2008
The earliest example I have been able to find is also from the New York Times, from late in 1884:
The general run of out-siders who put up dollars for theatrical enterprises very promptly show their right to the title of backer by backing out at the first symptom of loss. But now and then a plucky financier makes his appearance in this field. . . . Nobody knows how much solid cash Mr. Weed deposited in the venture. But he apparently had no regrets. It is customary for gentlemen like Mr. Weed and Mr. Williams to be alluded to by theatrical people as “angels.” Mr. Weed was called Frederick Bock's angel, Mr. Bock being his experienced partner in the various enterprises which cost all the money. By the same token Mr. Williams has enjoyed the celebrity of being generally alluded to as Mr. Tillotson's angel. That distinction certainly ought to be worth a part of the $36,000 Mr. Williams is said to have parted with since last Spring. Many capitalists would be glad to assure themselves of angelic surroundings at that price.
—"Manager, Actor and Play; A week at the theatres to give thanks for." The New York Times, Sunday, November 30, 1884 (unfortunately paywalled; if you aren't a subscriber, your local or institutional library may have access)
The assertion that it was "customary" to refer to investors as angels suggests that the term was not new in New York theatre circles even in 1884.
An early UK use and attribution can be found in an 1890 letter to the editor of The Scots Observer (emphasis added):
My newsagent informs me that Mr. George Moore has lately been in labour in the pages of a God-gifted paper called Short Cuts, with a proposal for a national subsidised theatre. All he wants is an Angel—(Mr. W. D. Howells's word for a millionaire capitalist, God bless him !)—with £10,000 a year to throw away.
Although this usage is in a publication from the British isles, "Mr. W. D. Howells" in this case almost certainly refers to William Dean Howells, described in Wikipedia as
an American realist novelist, literary critic, and playwright, nicknamed "The Dean of American Letters".
Howells spent the first half of the 1880s in England and Italy, just a few years before the Scots Observer article. And in fact, in one of Howells' own works, published in the US in 1889—a year before the letter above—we find this exchange (emphasis added):
“It will take a great deal to get such a thing going; and even if he's got an Angel behind him——”
She caught at the word: “An Angel?”
“It's what the theatrical people call a financial backer. . . .”
—William Dean Howells, A Hazard of New Fortunes, 1889
By 1894, the term was in wide enough use in the American theatrical world to be used to define other terms; in The Showman's Encyclopedia we find this entry (bolding original, italics added):
Sucker. An inexperienced backer of a theatrical enterprise; an angel; a grafter's victim.
—The Donaldson Guide . . . , in conjunction with The Showman's Encyclopedia . . . , and the International Professional Register . . . , to which is added The Complete Code of the Donaldson Cipher, W.H. Donaldson, Ed., Cincinatti, Ohio, U.S.A., 1894
Given the 1890 Scottish attribution to an American and the apparently commonplace use of the term by "theatrical" Americans by the late 1880s/early 1890s I think it is likely that this is indeed an originally American usage, probably coined somewhere in the 1870s or early 1880s (but possibly earlier), which very quickly made its way across the Atlantic, no doubt aided by the long history of intercourse and collaboration between "theatrical people" from the US and UK.