I am trying to understand the logic. If you want to make a case that the maintenance provider of a lathe has to be subjected to 8.4, then YOU MUST mandate that the manufacturer of the lathe is also subject to 8.4. Your logic DEMANDS that view, because the lathe is an "externally provided product" that can adversely affect the organization's ability to consistently deliver conforming product. Otherwise, there is no logic. You just devised a totally arbitrary line, to include maintenance providers, but not the manufacturer of the equipment that will be maintained, which, there is nothing wrong with it, but NOT SUPPORTED by 8.4, in my assessment.
The maintenance service of production equipment does not even 'touch" the intended product, so, I do not agree with your assessment that it falls under "ensure that externally provided processes, products and services do not adversely affect the organization’s ability to consistently deliver conforming products and services to its customers" .
A coating service, such as anodizing, for example, etc... TOUCHES THE PRODUCT, and directly affects product conformity. Not the same with a maintenance service.
So, just to be clear: there is nothing wrong in having your (production equipment) maintenance service providers subjected to 8.4, but in my assessment, it is not a requirement of ISO 9001, for the typical manufacturing organization.