Is it safe to clean vinyl with a wet washcloth? Also is this an effective method for cleaning vinyl? I’ve tried using anti-static vinyl cleaning brushes but they just seem to push the dust around rather than removing it.
Short answer: No.
You'll likely just spread dirt around with a wash cloth and/or rub dirt into the record.
For superficial cleaning use a brush designed to clean records. That can remove surface dust, hair, etc. Don't expect dirt or dust to come off on the brush; you'll want to sweep it off the record using a light touch.
For thorough cleaning, consider an enzyme based approach such as AIVS fluids with light agitation, or an ultrasonic cleaner. Either with distilled water for a rinse. There are other fluids and various machines, from manually operated to push button.
Cleaning vinyl records is a rather wide topic with lots of information and discussion available, here and elsewhere. Here's a couple links to a few of my own articles on the subject:
AIVS and other fluids:
http://www.theaudiobeat.com/equipment/aivs_products.htm
Ultrasonic:
https://www.whatsbestforum.com/threads/timas-diy-rcm.26013/
Thanks for your interest in cleaning records. A critical element in vinyl enjoyment is reducing noise from dirty records.