Is the time PHP takes to render a page and serve it to the client ever a problem?
Without finishing reading the body of your question, I have the perfect answer.
This is because all of the processing time required to present the page to the user from the moment the user hits the enter key (from typing in a URL or clicking a link) is of supreme importance. It's so important that if the amount of time is too long, the guest will think the site is broken.
The process time can worsen when alot of people are connecting to the server at once because the computer will have to handle each request along with the processing of each request and if there are way too many people connecting, then some people might experience delays before any part of their request is processed. This depends on the processing power of the server.
There is a term that is valuable to you. It's called "Time to First Byte". It means the amount of time it takes for people to see any part of the page from when they first request it. webpagetest.org can measure this value for you when you run any website through them. You want to keep the value as small as possible (roughly 20 to 50ms if pages are served locally, or up to 1ms if pages are served to a far away place with poor internet connection).
Now if you're looking for a way in which the time the PHP processor takes to render a page is meant to not be an issue, then you can either:
Ditch PHP and use another server side scripting language, but then you'll have to factor the processing time of that language.
Even if you take one of the above two options, you still will want to factor the time the server processes requests as a whole, even if it means the PHP and/or any other server scripting languages that are used to process the requests for the guests.