Can a pilot (other than an ATP) who is not certificated and current as a CFI give instruction in aircraft required for the issuance of a pilot certificate, e.g. a private pilot certificate?
I'm assuming that you're asking about FAA regulations. Anyone can train anyone else - how effective and safe that training is is a different question! - but only an instructor can issue endorsements, including signing off on training time, endorsing solo flight, and accepting a student pilot application (see 14 CFR 61.193 and 61.215).
This is from 14 CFR 61.51:
(h) Logging training time. (1) A person may log training time when that person receives training from an authorized instructor in an aircraft, flight simulator, or flight training device.
(2) The training time must be logged in a logbook and must:
(i) Be endorsed in a legible manner by the authorized instructor; and
(ii) Include a description of the training given, the length of the training lesson, and the authorized instructor's signature, certificate number, and certificate expiration date.
In theory, a non-CFI could train a student to perform as required by the ACS. But, a non-CFI couldn't authorize the student to fly solo, and he couldn't endorse the student's training time in his logbook. The student would end up able to fly but unable to get a certificate because he'd have no official training time and no solo time. So it wouldn't really be helping the student much: he'd have to go and do the required hours over again with a CFI anyway.