Question inspired by a thread at the moment on r/history. I've looked at previous threads but the only concrete recommendations I can find are the general book list (which is much more zoomed in on specific issues) and a post which has now been deleted!
Is there a good single-volume book that is both historically accurate and accessible to a general audience?
There isn't a book that will tell the exact same stories as Zinn, nor is there really anything like a less-biased history.
Of course bias isn't really the issue - all historians write with a bias, and the good ones try to be aware of that. The real issue with Zinn is that in his worse points, he makes broad claims that aren't really backed up properly with evidence. With that said, it's still the history of the US where I learned about Dorr's Rebellion and the New York Anti-Rent War.
If people are interested in reading history from a broadly Marxist background, then I would suggest substituting out Eric Foner for Zinn. Foner uses the tools Marxist historiography, but doesn't try to warp the historic narrative to fit a preconceived ideology. His masterwork is Reconstruction: America's Unfinished Revolution, which is monstrously huge and covers the 1865-1877 era. His Story of American Freedom is a one-volume history of the US which probably most closely approximates the materials and themes that Zinn addresses.
Still, if you want the most up-to-date one volume US history, Jill Lepore just released These Truths: A History of the United States. I haven't read it myself, but Lepore is a great historian and writer, and has covered a wide variety of topics in American history, from King Philip's War to Wonder Woman.
As far as an introductory general history of the US from a leftist perspective, I think this article from Current Affairs (“The Limits of Liberal History”) does a good job evaluating Lepore’s book. They find it rather lacking in labor history (Zinn’s bread and butter), and recommend Erik Loomis’ A History of America in Ten Strikes. Loomis is a well respected labor historian and I strongly recommend his book. It takes the working class’ (including slaves, women, and other traditionally marginalized groups) through American history to the present, using strikes to illustrate moments of continuity and change. I think it is extremely accessible the way you are looking for, it’s short (200 pages with big font, as compared to Lepore’s door stopper), and he gets a lot of mileage out of using, say, the Mill Girl strikes to talk about patriarchy and education in antebellum America.
Edit: A quote from the Current Affairs piece to get people to read it: [“ ‘Politics’ [as Lepore writes about it] is not about what happens in ordinary people’s workplaces, but about what the government does, and so Eleanor Roosevelt and Thurgood Marshall are significant while Jessie de la Cruz, Hattie Canty, and Luisa Moreno are not. I don’t think you really realize just how limited Lepore’s history is until you pick up a book documenting the excluded parts”] (https://www.currentaffairs.org/2018/10/the-limits-of-liberal-history)