The Aikido master you're talking about must be Gozo Shioda who fought a G.I. who used boxing techniques. He ignored the jab and entered deep, caught the dominant right hand and executed a shiho nage. He said afterwards that the jab was just a feint and that the real weapon was the right hand so he controlled it before it could do any damage.
According to him, you need to enter without being concerned by the opponent's attack because it will only result in confrontation. You need to move first because "clearly, you can't deal with a boxer's speed by waiting", and then rely on your senses to improvise with the best suited technique.
Source: Aikido Shugyo by Gozo Shioda.
Note: in the chapter that follows this story, he states that the throws he has most used in real fights to great effect were irimi nage, hiji ate kokyu nage (which is kind of the Yoshinkan equivalent of ude kime nage) and the fundamental shiho nage.
I've recently read a blog post stating that Osensei, when asked, used to answer that aiki has nothing to do with timing and I tend to agree.
I'm sure I don't understand everything about it but the aiki I've seen, felt and used to some extent happens even in static positions where timing is irrelevant (i.e. when a guy does a static grab on your wrist, the moment when you execute the technique doesn't matter). I might add that as long as you don't stop the flow, you can do it as slowly as you want and the technique will still work on a mechanical POV (of course you wouldn't want to slow down too much in a real fight because even though he's unbalanced you can't let the opponent take the initiative). In the video, for example, even if they were doing it slowly or static, he should be able to unbalance his partner with little effort without relying on the attack's momentum.
Like this (from 1:11 onwards):
The aikidoka in the "irimi vs punches" video seems to be actually "cheating" in a way: from what I see, he seems to be pulling the attacker's lower back and then come at his head with his arm, unbalancing his partner using both bodies' opposing momentums. And this sir seems to be using a lot of strength. Not to say that it doesn't work (even though I'd be curious to see this technique against a bigger opponent) but the mechanics at work are not the same as the aikido irimi nage.
It actually looks like a technique I've learnt in Kajukenbo, where you use your hip/leg to provide fulcrum, instead of blocking the lower back (see here at 3:41):
From what I can manage to grasp, aiki is the "blending" of your "force" with your partner's "force" by not opposing it, which allows you to subtly change the direction of both "forces" with a rotating/spiraling movement ("defensive" aiki). It leads you to a position/situation/dynamic where you can apply a throw, a pin or even a strike when your opponent is defenseless ("offensive" aiki). This requires very little strength. Using aiki, you can also connect your center to uke's for both the aforementioned defensive and offensive purposes. Aiki seems to be more of a structural concept and can be really static, as showed by this video (at 2:35):
Or here in Daito Ryu where "I apply aiki" does not mean "I use the opponent's momentum" (1:53):
To this, you can add the concepts of keeping a correct, relaxed and connected body structure (akin to what is studied in the Chinese internal martial arts).
Then, there is kokyu ryoku, which I understand as the "breath power" coming from keeping a relaxed and connected body and using the power of the whole body with the right timing for both defense and offense (there's an element of breathing too). Simply put, kokyu ryoku is the power one uses to move his center and uke's which are connected with aiki. It is more of a dynamic concept and includes timing: you need to blend with the attack at the right time: the better you blend and the stronger the attack, the more power you can redirect and add to your kokyu ryoku.
Those concepts overlap to some extent and my understanding of them is still very immature, yet I wanted to answer to the affirmation that aiki is "just timing".
Now going back to the original topic (sorry for the rant), I think that indeed some/too many flavours of aikido have become more of a "martial dance" than a martial art, focusing only on the "ki no nagare" (dynamic flow work) and throwing out of the window such basic concepts as good body structure, power generation, correct angles and lines, weapon work, resistance, etc.
Now the Founder was by all accounts a fantastic fighter and a lot of his students (and their students) were outstanding too. Even though the art has been diluted, there are still teachers left who teach aikido as a martial art. We just have to rediscover what made aikido work for OSensei and his students in fights. Moreover, a part of the aikido community seems aware of this issue and is working on solutions: we see more cross-training than before (just like the guy in the video) and some people are even training in internal arts to try and replicate OSensei's strength. I think that Aikido has a lot of tools that can make it really effective but it needs proper teaching, hard training and the courage to actually get out there and put the skills to the test, be it by messing around with like-minded people (I have done this and even though it didn't look like the kata and needed some shoulder-shoving I have managed to get sankyo, kotegaeshi, etc.) or by doing live-blade sparring in the dark like OSensei's students.
Oh and I think that if we figure out how to irimi against skilled resisting opponents a lot of problems will be solved.