I am about to finish my Ph.D. at the ripe old age of 34. I plan to stay on and try to become a part of the faculty at a university. Considering the time it takes, not to mention the competition involved, to become an assistant professor and then get tenure to become an associate professor and then finally a professor. What are the problems one might face because of completing a Ph.D. at an age quite older than most other candidates?
Your chances of "rising to the highest ranks" of academia are small. But no different from anyone else, no matter their age at completion of a doctorate. The bar is the difficulty of doing good, publishable, recognized research. True research is delving into the unknown. And it is, well, ... unknown.
Additionally, at the moment, the job market is tight and you need to get in the door. But once inside it is up to what you can accomplish, in your thirties or .... seventies (more).
Additionally there is a worldwide pandemic that you need to avoid if you are to have the time to progress. Plus all the other issues that might wind up resulting in your early demise (too many guns everywhere in US...).
But note that none of that has anything to do with age.
Get hired. Do good work. Stay healthy. Make a lot of friends. Work with your colleagues. Hope for some breakthrough. If it comes, it comes. If not, live a good life anyway.
Is it practically possible to rise to the highest ranks of academia in STEM in the US system after finishing a PhD at 34?
Definitely! If you mean ``highest rank'' as in Full Professor, it is definitely possible, and 34 for PhD is not late at all. I agree with Buffy's answer, only that I would add that in my experience there are many full professors who got their PhD at the age of ~35, and many in elite US universities. So 34 is not even exceptional in this case.