The nation’s outrageously high TV crime rate far outpaces the relative safety in which we all live. These make-believe murders — routinely committed, procedurally solved, reliant upon technologies that exist more in TV writers’ heads than practice — are one of the last sure-fire ways to get any notable network ratings.

But the always-appreciable Dana Delany does her level best to lift “Body of Proof,” a paint-by-numbers morgue drama premiering Tuesday night on ABC, from its stale premise and up to the level of a beautifully-flawed-character study, something more akin to Julianna Margulies’s legal procedural, “The Good Wife.”

Delany plays Megan Hunt, a talented neurosurgeon who was in a car wreck that left her with recurring numbness in her hand, making brain surgery all but impossible. She has now reinvented herself as a medical examiner — and a mean one, at that. “You can’t kill somebody once they’re already dead,” she snaps, bitterly.

This surliness extends to the rest of her life, where her ex-husband (Jeffrey Nordling) has limited her shared custody of their tweenage daughter, who also resents Megan for being a bad mommy.

Megan’s crime? Workaholism, for which she is truly sorry now: “I lost my child to my career and then I lost my career.” (Groan.) Yet she straps on those stiletto heels and stomps into her grisly business each day.

Delany brings a precision and cool beauty to the part along with a frowny, “House”-like, cruelty, which suits her. Unfortunately, “Body of Proof” is quickly overwhelmed in its first few episodes by a heavy caseload of dead joggers, social workers and chronic gamblers, the likes of which have all been murdered many times before on other shows, and whose cases must be solved in a tidy 42 minutes. Even with a confidently written female lead and an actor willing to give it her all, the crime-show formula is just too difficult to shake.

Megan’s co-workers (a well-assembled supporting cast, including Jeri Ryan as her boss, Windell D. Middlebrooks as another coroner, and Nicholas Bishop as a chiseled-chinned medical investigator) tread lightly around her, especially when she’s got a scalpel in hand.

A hurry-up pair of homicide detectives (John Carroll Lynch and Sonja Sohn) aren’t her greatest fans, either. Snapping on latex gloves and big-footing gracefully around the murder scenes in those heels, Megan is always about five steps ahead of the cops, ordering up magical labwork and fingering a suspect right away.

“Body of Proof” seems to treat as an afterthought its promising hope for deeper characters and longer story arcs, especially when it comes to Megan’s attempts to reconcile with her daughter. It leans too heavily on a trope (nobody likes a soulless career woman) and turns Megan’s efforts to warm up to her colleagues into a running gag about her social awkwardness. Instead, the emphasis remains firmly on the bankable simplicity of solving these crimes quickly and cleverly, with the requisite amount of wincing at rotting body parts and other viscera.

With the day’s stiff on the slab, Megan lapses into the gross anatomy’s idealistic “Quincy, M.E.”-speak: “I honor the body for what it tells me about [the murder victim’s] life and how that life came to an end. The body is the proof. It’ll tell you everything you need to know, if you just have the patience to look.”

Same with a good character. Delany’s Megan would take us places, over time, and reveal everything we need to know, if the people who make (and watch) these shows only had the patience for a fuller, well-paced story.

It’s too bad that such series have grown more tiresome in the past few years. I wonder what would we could discover if it were possible to cross-section the brains of people who are addicted to them — all this waste of character and writing over the years, all in service of a body count. Can there be a an answer? A cure?

But the minute we raise the knife and change the channel, the cadaver in the easy chair always snorts awake and pleads: Don’t turn that off, I was watching it.