Actually the "proper" name is "grease fitting" and it was designed by a guy whose last name was "Zerk" so calling it a Zerk fitting is primarily correct. The first company to mass produce it under Mr. Zerk's patent was the Alemite company, so "Alemite" fitting is the secondary identification of check-ball type grease fittings. Alemite is no longer the primary mfr'r of them so the more universal term "Zerk" is in wider use.
"Wonky" is British/Aussie/New Zealand slang for "misaligned" "crooked" or "awry" none of which mean "clogged" therefore improper adjectives for inoperable Zerks, even in American English, but maybe explains some things lately. (Where IS that tongue-in-cheek emoticon anyway...???)
Zerks which are "stuck" or "clogged" can be cleared with that tool you mention, but it's bad form because it forces the dirt-clog down into the bearing and damages it. That tool should only be used with the Zerk removed from the implement for clearing, then may be reinstalled.
But it may be simpler to throw it in the trash and install a new Zerk instead.
Zerks are made with SAE or Metric threads usually, both straight and taper or "pipe" and should be replaced with the same, so an assortment of Zerks might be the answer for the home handyman. (There rarely exist some un-threaded types which are press-fit but none in common useage.)
Horrible Fright Zerks are as good as any I've ever used and solve the problem nicely.
To keep them from clogging, one might consider what is common in aviation use... Zerk Caps, which protect them from dirt and water ingress.
https://www.amazon.com/gp/product/B07FXPK7SF/ref=ppx_yo_dt_b_asin_title_o05_s00?ie=UTF8&psc=1
