For training purposes, it is recommended that I get a tire with bead to bead puncture protection in the form of a Kevlar belt or just a tire with protection under the tread?

Fwiw, I've never had a sidewall blowout. I've never even had any nicks or gashes to the sidewalls of my tires. Or at least I've never seen any evidence of anything on the roads here tearing up the sidewalls of my tire.

Which do you recommend? I'm leaning toward the belt protection only because I like supple rides and also I like easier to mount tires!

Specifically, I am thinking about getting either the Kenda Kountach (tread protection only) or the Kenda Kaliente Pro (belt to belt protection).

Haven't used either of those tires but stiff sidewalls result in increased rolling resistance in addition to a harsher ride. So if you haven't been experiencing lots of problems with cut sidewalls I'd opt for a tire without extra stiffness there.

If your riding pavement or groomed trails, tread is sufficient. If you are riding wilderness where there are spike thorns, rocks, or anything that can possibly go through the sidewall, get bead to bead protection, also an air pump, spare tubes, and tire boots, so you don't have to push it back out.

Pay your money and take your chances.

Bicycle tire manufacturers generally offer a variety of models with varying degrees of flat protection. Riders make their tire selection, at least in part, on the degree of risk they are willing to tolerate. As a general rule more flat protection = more rolling resistance and a harsher ride so choose wisely.

FWIW, I've gone decades without ever completely wearing down a bicycle tire tread. I almost always cut a sidewall before the tread wears out. If I can still feel a bump after the tire has been repaired and re-inflated, I replace the tire.