I'm not an expert in this area, but I thought modern pistols and revolvers were safe to carry with a chambered round. Older single action pistols, the old cowboy guns, could discharge if dropped with a round in the chamber. My understanding is that modern guns have redundant safety features to prevent that.
Some modern pistols, eg Glocks, don't even have a safety, per se. They have a trigger interlock mechanism, where you have to press a secondary trigger before the main trigger engages. Typically police carry them with a round chambered. Military practice is to carry them with an empty chamber, at least according to Larry Vickers.
There are three distinct styles of firing mechanisms for pistols. One, the 1911 single action style, where you have to cock the hammer , either manually or by cycling the slide. They are typically carried "cocked and blocked", ie with a round chambered and the hammer cocked and safety on. Or they can be carried with an empty chamber, but since a single stack 1911 only holds 7 or 8 rounds, doing so costs you a round.
Two, are true double action pistols, which have a hammer which you can cycle by pulling the trigger, like a revolver. A Sig is a typical example. They usually have a safety and decocking lever. Typically, I believe you would carry these with a roudn in the chmaber.
Three, there are striker fired pistols, like Glocks, Taurus, etc. They don't have a hammer and are fired by an internal striker. It is a design that has certain advantages, such as avoiding the long trigger pull of the first shot of a double action design. They are designed to be carried with a chambered round.
I read that Zimmerman's kel-Tec didn't cylce after his first shot. That raised questions to me. Possible reasons for not cycling are a jam, or the slide being obstructed or held, suggesting a struggle for the weapon.
