The reason this book works, and it does, is that the author has a VERY good sense of what to take seriously and what should be handled with great silliness.
The premise is silly and Flynn clearly knows that. But the details of the early 19th century are elements that Flynn takes very seriously - time travel can be wibbly wobbly, but we need verisimilitude in our inns!
That's the thing - the balance is good. And Flynn conveniently writes from the viewpoint of a modern character, so all the first person reflections can use a contemporary voice and not fall into the trap of trying to be Jane Austen. Flynn only needs to get Austen's dialogue right, she doesn't need to get Austen's free indirect discourse. The latter is...really really hard, let's say.

Also, the main character is Jewish. For no reason other than that Jews exist and do things in the world so why shouldn't the main character be Jewish? And I appreciated that.