During the gunfight, Joe Lime fires six times without reloading, so in fact both he and Dillon had empty guns at that point.
When Matt gets back exhausted and sick, Chester asks if he got the brothers. Matt explains that because he was so sick he did not know but may have got one. He acknowledged shooting once. However when Chester went to clean Matt's gun he took out all six bullets from his gun. One should be missing.
When Chester wonders if Matt loaded his gun, the ammunition Chester unloaded from Dillon's gun is shown on the table. Seven rounds are shown and Dillon's gun only holds six.
Dillon is out on trail for two days without sleep and rides back into Dodge with a fever and can hardly walk and yet is still clean-shaven.
Chester only unloaded six rounds onto the desk (seven rounds are shown later). They all sounded the same. An empty one would sound different.
Matt carries extra bullets so he could have easily replaced the spent round.
As an experienced shooter, Dillon would have realized the gun was empty (too light) as soon as he pulled it from his holster. He would have reloaded it from the ammunition he carried on his belt, so the ending could not have played out as it did.
The details of how and when Matt happened to go after the brothers is never really revealed. It's possible it was lost to editing for commercials.
Knowing he had a fever, Kitty wakes up a sleeping Dillon, then tries to prevent him from confronting the surviving Lime brother. If she were concerned about Dillon, she would have left him sleeping and summoned someone else to help.
Doc refers to "old man Carter and that boy" as the murder victims. But the actor portraying Doc is 14 years older than Carter and looks it.