We assume that Katniss doesn't convince everyone in the districts of the authenticity of her love for Peeta because Snow says so. Snow believes that convincing the districts will calm them down. But Snow, as characterized in the prequels, fundamentally does not understand love.
Katniss DOES convince most everyone in the districts. The only district people to question the love story are the other victors, because they alone understand how the Capitol's propaganda works, how stories are controlled and victors are forced to play their part.
But when Katniss visits District 8, everyone seems fully convinced. People assure Katniss that they know Peeta was just speaking under duress. They're so upset about Katniss "losing the baby" that one woman actually weeps over it. The everyday District citizens buy into it fully.
The issue is that Snow doesn't understand how powerful love is. Katniss and Peeta's act fuels the rebellion instead of tamping it because choosing love over violence is rebellious. Like when Katniss and Peeta hold hands and Haymitch calls it the perfect touch of rebellion, or when Katniss covers Rue with flowers, or when all the victors hold hands on interview night. Compassion, love, friendship between competitors, it's all in defiance to the Capitol's agenda.
Snow thought that the districts would hear, "I'm just a silly little harmless teenage girl doing impulsive things because I'm in love! How nice of the Capitol to let me keep my boyfriend!" Instead they heard, "My love gave me the courage to defy the Capitol." They heard, "My love held firm in the face of all the Capitol's power and cruelty. I refuse to kill him or even passively let him die." They heard, "The Capitol is not strong enough to corrupt a steadfast heart." And if the Capitol can't even overcome two teenagers' puppy love, it certainly can't overcome thousands of adults' love for their neighbors, children, and homes, or a unified people's passion for freedom and justice.
Snow underestimated love, and it was his downfall.