I have some issues with thinking of inheritance because it implies some exclusion. To me that seems like an incongruence in the teachings versus universal salvation. So how to see beyond that?
In theory, if you remove all of the moral implications, the opposite of humility is just myopia or tunnel vision. Where one is so attached as not being able to see the larger picture. So that blindness leads to a loss of control and loss of power to be an apt custodian. That would one way to look at it with logic. Yes I think the quiet mind opens up the space and helps with this issue.
I think the term "inheritance" means more than some entitlement or reward to the chosen ones. Instead I interpret as a deeper message addressing our connection with life. To our true source referred to as the Father. So as you submit to the source you gain everything. And as you realign with the source you regain everything. Then humility becomes recognizing when you are out of alignment, and admitting you are out of alignment (submission/forgiveness).
And if you look at "the meek" as an action, rather than some static population of people, then it includes all of those who find it within themselves to turn their hearts. So this inheritance is available to all when they become ready to realign. As each heart turns it makes the journey easier for the rest. A collective transformation.
To me the absolute master of humility + peace = power from the last century is Morihei Ueshiba the founder of Aikido. Take a look at his teachings and I think you will find they are happily aligned with Jesus' teachings. Sure enough here is a quote that addresses your questions directly:
If your heart is large enough to envelop your adversaries, you can see right through them and avoid their attacks. And once you envelop them, you will be able to guide them along the path indicated to you by heaven and earth.
Morihei Ueshiba