John, I’m afraid you’re basically right to conclude there isn’t a great one-volume Burke biography out there. It’s sad to say, but the two top contenders were published in 1887 and 1927, respectively. They are John Morley’s Burke and Bertram Newman’s Edmund Burke. Morley tells the basic story of Burke’s life quite wonderfully, though his firm disapproval of Burke’s opposition to the French Revolution distorts some of the later portions; and Newman does a fine job of the basic narrative and generally weaves Burke’s views quite nicely with his life story. Both are of course long out of print, but are widely available at used bookstores and online, and both are well worthwhile. But frankly all biographies of Burke written before his immense body of personal correspondence became available to scholars in 1948 are sorely lacking, and there just hasn’t been a good one-volume biography written since that time. A few scholars have tried (most notably, in my opinion, Stanley Ayling, and C.B. Macpherson), and as you say there are many academic books about Burke’s political thought, but no one has really written a straightforward single volume biography for the serious lay reader. It is a niche worth filling.
For an excellent very brief biography, and a fantastic overview of Burke’s political thought too, I think the best place to look is Harvey Mansfield’s introductory essay to his Selected Letters of Edmund Burke. The introduction is well worth the price of the book, and of course the letters themselves are wonderfully interesting (and the collection is much more manageable than the complete ten volume Burke correspondence, which costs a gazillion dollars).