GFCI Did Not Trip - Outdoor Extension Cord Fell in Mud

It is possible that the GFCI device did not function as designed. Impossible to say without forensic analysis (which is kind of hard for random people on the internet). But it is quite likely that it did not fail:

GFCI protects exactly one thing - an imbalance between hot and neutral (on a 120V circuit) or between the two hots & neutral (on a 240V circuit). It does nothing to limit the total flow of current, as long as all of the current going through hot also goes through neutral.

A device on an ordinary circuit is designed to use a certain amount of power (current & voltage) and is designed so that under normal circumstances the heat generated will not cause any problems. But that doesn't mean problems can't happen. For example, a rolled up extension cord can easily overheat, even when used at the design (for unrolled) specifications. A toaster run in an enclosed space can overheat and easily start a fire. A heavy load (e.g., a full 15A) run for an extended period of time on a thin extension cord (perhaps designed & rated for max. 10A) can overheat the extension cord - causing it to melt - without tripping the breaker because the current is less than the breaker rating. In many of these cases, a regular breaker will trip once things start melting because then there is a short circuit == high current flow. In some cases - e.g., a 3-wire cord where the melting happens first between the ground wire & hot or neutral instead of between hot & neutral, a GFCI will trip.

Back to the specific example:

If everything is sealed sufficiently that, despite the outside being wet, no current is able to "escape", then the GFCI will not trip. However, overheating will gradually happen because mud is an insulator, leading to melting, short circuit and a regular breaker trip.

GFCIs are not magic everything-fixers

It's like asking an ambulance crew "why didn't you arrest those graffiti artists?" "That's not what we do".

GFCIs do one specific task that is related to human safety, and they do it well. They detect current leakage between the intended hot-neutral loop and anywhere else. However they do nothing about any fault that stays inside the hot-neutral loop.

Hence their name, "Ground Fault Circuit Interruptor", which is ironic since they don't interact with ground at all. It's presumed that the "anywhere else" must be ground.

Those other bailiwicks

It sounds like you had more of a "poor connection/arcing/overheat" problem. That is more in the bailiwick of an AFCI. (Arc Fault Circuit interruptor). Those work by listening to the power line for that familiar "crinkle-crunch" sound of hooking up speakers with the amplifier turned on, or a shoddy headphone jack, etc.

You note also that the GFCI did not trip and the breaker did. That's another thing GFCIs aren't. Some people think if they put a 15A GFCI on a 20A circuit, they can extend from there with 15A wire and the GFCI will protect it. The opposite is true, 15A GFCIs are specifically rated for 20A pass-through. (because of a NEC technicality).