Customized Resumes w/ Talent Stack Mapping
The high specialization makes it difficult for people not familiar with the specialized skill to understand how it may apply to the job they have open. Therefore, you will have to call that out on each job application / resume for them, which puts a more work on you.
Take your example of a person that operates a machine of which, there are only three in the world. Let's call these machines, "Machine Foo". Let's say the job you are applying for uses "XZ Machines" which are not as rare, but still expensive.
A person must have some super high level of skill - and a personality - that generates enough trust to let them touch a Machine Foo, let alone operate one.
Those skills and personality traits can be generalized into Talents that everyone would want in an employee, regardless of technology: Technical mastery, dedication, ability to train and work with others, diligence, etc.
The uniqueness of the machine makes these skills far more important than in other industries and that makes it easier to back up your claims of being great in these Talents because management is trusting you with very expensive equipment.
Machine Foo Operator
Although Machine Foos are almost unique (only three in the world!), they are very similar to your Company's XZ Machines in design and operating methods, and I will be able to easily map my successes below with Machine Foo to your machines.
Demonstrated technical dedication and appropriate prudence to be selected to operate Machine Foo - of which there are only three in the world. These machines, due to their rarity are worth $X,XXX,XXX each.
Management placed their trust in my skills, foresight and planning to hand over the machine for me to operate entirely.
I am one of only two trusted inspectors for Machine Foo. I monitor the health of the machine to ensure any issues are discovered before causing any major damage.
I supervise and train all others using the machine to ensure that the machine is well kept and properly handled.
Due to my continued monitoring and diligence, the machine has never broken down, never needed a major repair and is running at 99.9% of its productivity potential.
Because Machine Foo is similar to other machines at my company, I have been asked on several occasions to help set up, monitor and operate these machines as well because of the trust management places in my skills, troubleshooting and diligence. Once they are properly running, I then train other operators on their use and handling. Some of these machines were XZ Machines.
By showing personal characteristics (dedication, trustworthiness, mastery, etc) and showing how Machine Foo maps to the XZ Machine (your target), you should be able to compete with just about anyone. And, given that Machine Foo is more expensive and rare, the fact that you were trusted with it, makes you a better candidate to handle XZ Machines.