You can begin a sentence with but—no ifs, ands, or buts about it. I discussed this point in a 2018 post and readdress the issue here with examples from writers far better than most of us who began a sentence with the conjunction.
"But with defeat comes the need to talk, to explain, which presented problems for a fiction writer." Joseph Blotner, Faulkner: A Biography, Volume 1 (1974), page 691
"But it doesn't matter with me now." Rev. Martin Luther King, Jr., "I See the Promised Land" 1968 speech, in A Testament of Hope: The Essential Writing and Speeches of Martin Luther King, Jr. (1986), page 286
"But in 1861 these achievements still lay in the future." James McPherson, Battle Cry of Freedom: The Civil War Era (1988), page 321
"But the atmosphere of crisis that has been generated by the confusion among some of the white Americans who control the 'education establishment' is cause not only for alarm but for red alert and hot pursuit." Albert Murray, The Omni-Americans in Albert Murray: Collected Essays & Memoirs (1970), page 177
"But there's a problem, which has more to do with space than race: The longer form does not lend itself to headlines." William Safire, Watching My Language: Adventures in the Word Trade (1997), page 110
These five examples are from five different stylists: a biographer, a social activist, a historian, a critic, and a rhetorician/grammarian. I can offer many more examples of accomplished writers from other fields using but at the beginning of a sentence with their editors' approval. The point of raising this issue is not to encourage starting sentences with but. Rather, I want to reinforce the point that rhetorical rules are usually arbitrary, especially when cited by writing teachers who rarely write. True, overusing but to lead off a sentence may suggest a weak writing style. But the writer's goal is to employ a variety of sentence beginnings, middles, and endings. How can we create this effect without turning to all options available to us?