It looks like the rectus abdominis plays a role in posture and
contributes a bit towards increasing intra-abdominal pressure (IAP).
Posture is more dynamic and complicated than people give it credit for. Technically every muscle in your body plays some role in posture.
Also, it is used for trunk/lumbar flexion as well.
Of course, it does that but priority of training that is likely low.
I'd say the torso is more for transfer of force from the upper body to lower body and vice versa. Secondary or tertiary function is stability, or anti-rotation (could be considered the same thing). As opposed to lumbar flexion, which is at best a tertiary or quaternary (fourth) consideration.
Frankly few people are that weak at lumbar flexion. And if you can do more than 8 reps of something, you're no longer training strength, you're training muscular endurance. Doing 30-40 crunches becomes a waste of time because the threshold of adaptation is way too low and it takes to long to get a training effect at the expense of spinal flexion cycles.
However, what benefit is there to having a particularly "strong"
rectus abdominis? By strong, I mean significantly beyond whatever may
be required for basic mere things like posture and the occasional
sit-up (or crunch).
- Hypertrophy -- If that's what you want, then loaded lumbar flexion (cable crunches, loaded crunches) is somewhat of a requirement in a ~6-12 rep range.
- Function -- Especially explosive anti-rotation work like pallofs or chops/lifts or in eccentric anti-flexion like a roll-out. These are all particularly useful benefits for athletes, especially rotational or overhead athletes as it contributes to force generation/transfer.
Yes, none of those examples seem 'specific' to rectus abdominis in terms of isolation, but you can't isolate muscles anyway. This is a myth, you can only favour muscles in a kinetic chain. All muscles work in some kind of kinetic chain. Even in a crunch, your hip flexors, TVA, pelvic floor and your obliques are involved.
The RA is involved heavily in all of those torso exercises.
I understand IAP can be quite helpful for lifting and handling heavy
weights, but it seems it's the transverse abdominis that's much more
important for that.
I'd argue that internal obliques and your diaphragm are more important than TVA for most people for strong IAP. Breathing and bracing drills improve IPA significantly better than TVA specific drills (vacuum/sucking in). In many ways, TVA specific training is the opposite of creating good IPA. TVA specific work is overrated and based on generally outdated research.
For example, is there any "heavy lifting" or exercise significantly
helped by a specifically strong rectus (not transverse!) abdominis?
I'm not referring to some contrived "weighted sit-up" or crunch
Yes. The chin up immediately comes to mind as you have to control extension. Anyone good at them will typically have considerable RA strength. There is a piece of research or two, that showed chin ups yield significantly more RA activation than any crunch variation (even loaded ones if I recall).
Of course, given that you can't truly isolate one muscle and that all muscles work in some kind of kinetic chain. It's silly to really focus too much attention on any one muscle.
There are also numerous athletic reasons to have a strong RA but not traditionally as you might think of it (i.e. crunch). Explosive rotational power needs a strong RA to slow rotation, so every pitcher, every tennis player, every volleyball player, etc... All need strong RA's (in conjunction with other torso muscles) to prevent injury.