Ultimately it comes down to this. Without Intimidate it will be very difficult to counter the threats I mentioned.
Infernape
Here's +2 Infernape Close Combat against max/max+ CAP10: 75.5% - 89.1%. We're looking at a small chance of ohko with just stealth rock, and a very high chance of ohko with stealth rock and spikes, or two SR switch-ins. Note that Ape can just switch out and come back in again later, at which point you can no longer counter it.
I realize that our definition of "counter" is a bit vague, but it probably shouldn't be "has to be at complete full health to have even a chance of surviving an SR-neutral Poke's attacks and dies if even minor residual damage is present".
To whoever suggested Aqua Jet, here's Aqua Jet damage from two spreads, the first being from the same spread above, and the second being from a max ATK spread (assuming no boosting item). This is against 4/0 Ape.
0 atk CAP10: 34.7% - 40.8%
Max ATK CAP10: 49% - 57.8%
So no, Aqua Jet isn't helping anything.
Tyranitar
Assuming max/max+ CAP10 again, CBTar does enough damage with Earthquake and Crunch that it'll prevent us from stalling it out with Recover (when taking the secondary effect of Crunch into consideration). This isn't such a big deal if we assume CAP10 gets Superpower or something, since we'll always outspeed Ttar, but it is a big deal if our best attack against it is going to be Waterfall.
Against DDTar we have much bigger problems. It's always going to outrun and ohko with Earthquake, and we won't outrun it unless we have a scarf (and if we have a scarf and it happens to be a CBTar we're up against, Earthquake nearly ohkos while all its other moves do massive damage).
Dragonite
Max/Max+ spreads simply lose against the Dragon Dancer. Scarfers can outspeed and ohko of course, but again, if you end up taking an Earthquake on the switch you're not going to be very happy. And before anyone says "well Dnite never spams EQ!", it sure as hell will start if CAP10 is running rampant.
There are others (some of which I mentioned in my previous post but don't want to do calcs for right now), but my general point is this. There are multiple, extremely dangerous physical threats that our "Utility Counter" will vaguely check at best if it doesn't have Intimidate. It's just that simple. If it's a part of our concept that we cannot win against offensive threats that many teams have issues with, then fine, don't vote for Intimidate. But if that's the case I guess I just understood the concept wrong.
Edit: For those suggesting Shuca Berry, here's +1 Outrage against 0/252 CAP10: 93.7% - 110.4%. If Mence sees your lack of an item, it's obviously going to Outrage instead of risking EQ (which does only slightly more damage). And you need to be 0/252, because if you try to run max/max+, Mence beats you just by EQing/Outraging on the switch.
Of course both Intimidate and Trace help against this by knocking down that +1...but the same can't be said for Dragonite!