If they're rich, they should be able to easily solve this problem as soon as it starts causing trouble for them, by paying the workers much more money.
The article says doctors at the local hospital making offers on million dollar places are getting outgunned by out-of-towners making cash offers. It seems optimistic to think that anyone is going to pay e.g. someone working at the deli counter in the grocery store enough to afford the cost of living in the area.
And if somehow there is a push to increase minimum wage in the town to help with that, I guarantee I can tell you what will happen next: landlords will increase rents to slurp that into their accounts, leaving them with the same problem.
By definition, as long as there are more prospective renters than actual housing units available, prospective renters will compete with each other spontaneously by bidding up their offers. That's true everywhere.
That's just how competition and having multiple people submitting multiple offers work... the best offer is accepted.