Starting each chapter with a quote is a frustrating trend among writers. I see multiple manuscripts every day do this.
I never read the quotes. I’m reading to get a sense of the writing style and plotting capability. I don’t care about some pretty words the author dreamed up to give voice to historical backstory. Or words someone else penned.
Some authors take quotes to the extreme. Three big quotes at the start of the chapter took up an author’s entire first page. A memoirist quoted themselves beneath each chapter heading: trite, simplistic sayings for living well. They may have meant well, but they came off stuck up.
Having just short, one-sentence quotes beneath each heading can still be an infraction.
First guideline:
Remove quotes from the beginnings of your chapters. You don’t need them.
Most readers will skip over them. If there’s pertinent info, move it to the body text so readers don’t miss it. If there isn’t, why use the quote?
Books aren’t about being pretty. They should be pretty, but it isn’t what they’re about.
Second guideline:
Ask yourself: Does this move the plot along? Does this give someone voiceless a voice, and why should that occur here instead of in the narrative?
A epistolary-hybrid novel might start each chapter with a journal entry. The entry could be the protagonist’s or someone else’s. Letters or newspaper clippings also work. In this case, keep it short.
As always, you may break the rules if you do it well enough. A friend of mine interwove bits of a Korean fairytale between chapters. It turns out the fairytale is behind everything happening.
DUNE has snippets of religious texts starting each chapter. Things readers need to know but characters won’t address, or tantalizing reflections from the future. Herbert wanted to show how shifts in power change people’s perceptions of historical figures.
Third guideline:
Consider removing the quotes while querying. If the agent wants 5-10 pages, you’re only handing them a piece of Chapter One anyway. Dispel with the quotation and include another paragraph or two of narrative.
I don’t read quotes, but narrative is make-or-break.
In the Inbox 
That trend seemed to be huge back when I read Christian fiction (CBA historical fiction series, mainly), which was in the early-to-mid 90s. I, too, always skipped those quotes. Sometimes they added to the story but sometimes they didn’t.
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I’m with you! I used to read quotes religiously because I read every. single. word. of the book. These days I find myself skimming the quotes and not retaining a word of them afterward.
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