The three suggestions posted as I write this—"rip off the Band-aid," "grasp the nettle" and "take the bull by the horns"—focus on the (relative) desirability of confronting a problem or enduring a hardship directly and getting it over with quickly. But the anecdote that the poster asks about focuses on the opposite point: the (relative) undesirability of confronting a problem or enduring a hardship indirectly or slowly. A further element of the original anecdote is the idea that the person inflicting the suffering does so out of imagined kindness. It is rather as if we had a proverbial phrase in English about removing someone's nose with a grindstone instead of a sword stroke in order to be more humane.
I can think of two expressions in English that emphasize slow infliction of harm. The older one appears in John Ray, A Collection of English Proverbs (1678):
One had as good be nibled to death by ducks, or pecked to death by a hen.
Although this expression is usually shortened to "nibbled to death by ducks" or simply "nibbled to death," it also appears in variant forms. For example, from "Speech of Hon. John Kasson of Iowa in the House of Representatives, Saturday, May 6, 1882" (1882):
Mr KASSON. ... Repeatedly has the effort been made for a revision of the tariff, and I can sum up the result by quoting the declaration of a former chairman of the Committee on Ways and Means, who said that he found that the bill under his charge “had been nibbled to death by pismires [that is, ants] and kicked to death by grasshoppers."
The newer (but still fairly old) expression evidently reached English from a Chinese legal term, ling-chi. From "Abolition of Torture in Chinese Trials," in The Interpreter (June 1905):
At least we know of one form of cruel death which has been forbidden, and this is that which is technically called Ling-chi, or death by a thousand cuts ; each cut designed to produce torture, but carefully calculated to avoid a prematurely fatal issue.
And from Lyman Cotten, "Why China Sleeps," in The Arena (August 1909):
A party of tourists were being shown through one of the largest cities in China by a guide well versed in the English language. Upon reaching the public execution grounds he explained, with the fluency and vividness as to gruesome details of the usual guide, the different methods of execution in vogue, from simple beheading to slow strangulation and death by a thousand cuts. One of the ladies of the party, much impressed, and, with horror in her voice, exclaimed: "But you don't do such things now?" "Oh no" replied the guide with slightly puzzled manner, "not now: four o'clock."
The expression appears figuratively in English at least as early as "How It's to Be Done," in The Review of Reviews for Australasia (September 20, 1903):
Mr. Tom Mann says he would pile taxes on the land-holder till the guilty wretch is glad to hand his land back to the State ; but this is confiscation in instalments. It is the Chinese punishment of li-kin—death by a thousand cuts—translated into commercial terms. The other Labour leaders dismiss the problem with the reassuring comment that, * Thank God! it's a long way off.
And it has appeared figuratively without explicit reference to the Chinese legal punishment since at least 1913. From Ian Hay, Happy-Go-Lucky (1913):
Willing hands dumped the mummified and inanimate form of Jebson [who had been bound with table napkins and rolled up in a tablecloth] into an armchair, and the unique collection of Sports sat round him in a ring.
Then suddenly Dicky lughed.
That's all, Jebson," he said. We are n't going to do anything else with you. You are not worth it."
Mr. Jebson, who had been expecting the Death by a Thousand Cuts at the very least, merely gaped like a stranded carp. He was utterly demoralised. To a coward, fear of pain is worse than pain itself.
Figurative use of the expression seems to have been especially popular in the first third of the twentieth century in political debates. Thus, for example, we have instances such as this one in remarks by Sir John Simon, in Parliamentary Debates (Hansard) Official Report (1931) [combined snippets]:
It is no secret if I say that I understand there is some slight difference of view on these benches as to what the Liberal Party are going to do. I myself am a supporter of humane slaughter. My hon. and learned Friend the Member of East Nottingham seems to prefer death by a thousand cuts. But, of course, there is no misunderstanding between us, because, as a matter of fact, the Government do quite understand that, of course, their Bill is never going to pass.
The two expressions that I have suggested lack the element of intended kindness that makes the poster's anecdote so striking, but otherwise they seem reasonably on point. The only idiom I can think of that hints at the mistaken benevolence component of the poster's anecdote is "killing [someone or something] with kindness." Christine Ammer, The American Heritage Dictionary of Idioms, second edition (2013) has the following entry for that expression:
kill with kindness Overwhelm or harm someone with mistaken or excessive benevolence. For example, Aunt Mary constantly sends Jan chocolates and cake and other goodies, even though she's been told Jane's on a diet—nothing like killing with kindness. This expression originated as kill with kindness as fond apes do their young (presumably crushing them to death in a hug) and was a proverb by the mid-1500s.
The drawback of this expression is that it suggests complete obliviousness on the "killer's" part to causing harm or pain, whereas the man chopping off his dog's tail by degrees knows that the operation will hurt the dog but imagines that starting at the tip of the tail and chopping his way toward the base will somehow make the pain more endurable.