Electability is an amorphous concept that can be a disadvantage to those who don’t fit the mold of all but one past president: white men.

“The people who want a strong male leader who acts like a football coach are probably never going to vote for her,” said Nancy Gaub, 65, from Fairfield, Iowa, who attended a Warren event in December.

On Tuesday, Ms. Warren came to the Des Moines debate armed with a fresh answer for her critics. “Look at the men on this stage,” she said. “Collectively, they have lost 10 elections. The only people on this stage who have won every single election that they’ve been in are the women.” She went on to note that she alone had beaten an incumbent Republican in the last three decades.

Her pitch as a unifier, however, may have suffered in the moments after the debate when cameras caught her apparently refusing to shake Mr. Sanders’s outstretched hand. Then on Wednesday, audio surfaced of that encounter, with Ms. Warren and Mr. Sanders trading accusations that each had called the other a “liar.”

Longtime friends and allies, Mr. Sanders and Ms. Warren have sparred in recent days over whether he told her in a private meeting in late 2018 that he did not believe a woman could be elected president. She said he did make the remark; he has denied it.

Running as a “unity candidate” has also left Ms. Warren sandwiched between two politicians in Mr. Sanders (on the left) and Mr. Biden (to her right) with durable bases of support.

And though Ms. Warren is known for her plans, her decision in November to release her own sprawling “Medicare for all” proposal has particularly damaged her standing with moderates who balk at the plan’s $20.5 trillion cost and resist the idea that it could cause them to lose their private insurance, according to interviews with voters.