Thanks for all your comments on this.
If your system is so imbalanced as to produce enough radial force to damage the mount, you are overloading the motor ball-bearings massively.
Agreed.
You can't get away with grossly imbalanced high speed rotor.
The imbalance will only be existing in the process of outbalancing a new wheel.
I presume you are using wheels at much higher rpm than they are designed for.
Definitely
The wheel I use is is from my children Lego parts they do not use anymore (youngest is 17yo).
This can even lead to violent mechanical explosion if the wheel isn't strong enough.
That is the reason why I do the experiments in my Motor Test Station (MTS) which is surrounded by safety glass. I have seen complete destruction in the past 
Which wheels are you trying to spin at what speed?
Lego 13x24 wheel and Lego 13x24 tire.
What are you trying to do that you have to spin the wheels that fast?
Try to see whether my A2212 outrunner motors can show new speed record in Motor Test Station:
https://forum.arduino.cc/index.php?topic=331722
The thread started with slow speeds and people telling me that gears are a must. I never used gears in that thread and superglued wheels directly onto motor shaft. It ended with speed record of 17.79m/s or 64.04km/h or 39.80mph sofar, with 24mm diameter Lego wheels. Lately I measure rpm/rps by 90fps Raspberry camera slowmo videos from above MTS (instead of fork type infrared speed sensor), here is the record run:
For the forces heard and seen with A2212 2700KV outrunner motors I will not get away with lightweight wooden beam in motor test station as in the past. I already built a (159g) 315x34x34mm wooden beam, that is in place on MTS vertical center nail already. The 34x34mm will allow to fixate the 30x30mm aluminum mounting crosses:

As an engineer I can see your problem is all down to your wheel being out of balance. The plastic tyre on the rim is quite heavy compared to the rim, and it would only need to be a fraction of a gram heavier on one side to produce your problem of vibration. If at all possible - find another type of wheel if you need it to go at that speed,
Will do if I cannot outbalance the wheels. As you can see here I got the right wheel perfectly outbalanced already, just need to get a 2nd wheel that is outbalanced as well.

otherwise you can use a gearbox to reduce the speeds.
That is not an option, at least for new speed record.
Find a wheel that is totally machined and not cast or moulded as that is likely to be better balanced.
Will search for that. On the other hand the previos speed record was done with 24mm diameter Lego wheel with tire superglued on wheel superglued on motor shaft, with more than 233rps/14000rpm at motor shaft.
The aluminium brackets are failing because the holes in them form a weak spot, and the flexing caused by the vibration first hardens the metal and then cracks it.
I think I will be able to use aluminum mounting for outbalanced wheels.
With using wood it can flex without work hardening so is actually pretty good for motor mounts because it is easy to work, add holes and shape.
So wood is what helps in the process of outbalancing a wheel.
Hermann.