It doesn't have to do with the rules. The duration of impact(dwell time, FFS) typically encountered in table tennis just won't excite the modes that are anywhere close to 5000Hz. But with a suitable impact hammer that can produce a broad bandwidth, you can excite ALL of the modes. Therefore, even a premade can be a "5000Hz blade." The dwell time is the deciding factor here.
He's obviously talking about a blade having a primarily 5000Hz response on a sharp impact.
It's a fundamental property of the blade shape and material (well the blade/rubber/sponge system since they are attached). Also technically depends on where/how it's being held/clamped.
So in this case dwell time, and a suitable hammer are irrelevant. You might be able to get it to vibrate at 5000Hz off a single impact, but it will always vibrate at whatever it's dominant mode(s) is more. You arn't going to excite a primarily 5kHz vibration in the bat short of literally just shaking the thing at 5kHz.
The other modes of mechanical resonance are really only going to be dominant when you have excitation at a frequency near it's other modes, so are only relevant when your excitation is periodic. For a single impact perturbation, you just get the object mostly vibrating at the primary mode.
This brings us to the other problem. All these frequencies (at least of the blade), even the natural frequency, are largely irrelevant to the issue of speed.
Mechanical resonance is a thing. But it's primarily about energy storage. This is why it's relevant when forces either happen periodically, over a long time (eg Tacoma Narrows bridge), or when there is more than a single impact event, and energy from a previous impact(s) is still stored in the system before the final impact (eg Angers bridge, springboard or trampoline).
We can obviously rule out the energy storage component, as table tennis is about single impacts between bat and ball where all stored energy in the bat is released or wasted before next impact.
Which leaves us with the part of mismatched phases. This seems relevant at first, eg if you stand on a springboard until you and the springboard are both at rest, then jump off it once, you will get significantly less height than if you jumped off the ground because the springboard will absorb some of the energy of you pushing off.
The problem being, given that you are not allowed to preprime your springboard for this hypothetical (since you don't get to impact that table tennis ball twice) and therefore cannot use the springboard to actually store useful energy. Then in fact any impact is modeled by jumping from rest, on a springboard-person system at rest.
As you apply force in your jump, the force goes to deforming the board further, until the point that the restitutive force is equal to the force you apply pushing on the board, at which point the board cannot deform any further, and you start to push off, once you build enough velocity relative to the now stationary board, you lift into the air, after which point the board no longer affects you.
As you can see, in this scenario, phase matching is impossible. At the point of departure, the board is in fact acting as a stationary platform. The faster the resonance of the board (or the more resistant it is to deformation) the sooner you lift off it but it is still acting as a stationary platform at the point of lifting off, irrespective of how long/far into it's deformation it takes to get to this point. The only thing that changes is how much of your energy of your jump it has wasted to get to that point. There is nothing you can do in terms of timing to make this jump more efficient, the board is passively waiting there for you to complete your jump.
So in the end, the frequency of a springboard/blade is pretty much irrelevant. You can't actually phase match 2 objects that have a speed but are otherwise passive in a collision. All you can do is make sure that the most elastically efficient components take most of the deformation.