Reduced relative clauses with present perfect forms

The active/passive dichotomy may not be helpful. Arguably the reduced relative clauses occur more frequently in spoken rather than written usage, where - to use your example - "The pipe that has been broken will be repaired in a week" would be a stilted, or at least unusual, form in spoken English.

The example sentences seem grammatical, but I don't know if it's right to think of them as containing reduced relative clauses derived from active present perfect forms.

"The system implementing the new software is functioning well" is not wrong, but I can only read it as a reduced version of "The system that is implementing the new software is functioning well," not as a reduced version of "The system that has implemented the new software is functioning well."

"Applicants passing the exam go through to the interview" also sounds OK to me, but I don't know how to explain its structure. I'm not happy with calling it a reduction of "Applicants who are passing the exam..." because that actually sounds odd to me, but I'm also not convinced that it's particularly closely related to "Applicants who have passed the exam...". To me it seems closest in meaning to "Applicants who pass the exam go through to the interview."

Perhaps it can be compared to sentences like "Some items belonging to customers were stolen." Obviously, we would use a non-progressive form in a relative clause with a finite verb: "Some items that belonged to customers" is better than "some items that were belonging to customers".

Please correct me if I’m reading your question wrong here, but to me it seems that the crux of the confusion stems from the seemingly a-grammatical nature or meaning shift in changing:

Applicants who have passed the exam go through to the interview.

To

Applicants passing the exam go through to the interview.

I would like to point out a feature of the perfective that appears to have been forgotten in the other answers. The perfective helping verb have can also undergo mutation to fit the grammatical needs of a sentence. Therefore, I propose that the reduced relative clause most matching your original sentence would be:

Applicants having passed the exam go through to the interview.

The question of whether this would be used more in speech or in writing I believe is a secondary, though equally important question, though outside the scope of this post.