Most lubes have a solvent carrier. That will clean while you apply and rub.

That's all I ever bother doing.

If I do pull a chain I have tall "Ball" canning jars that I fill with order less mineral spirits. Put the chain in, shake, pull chain out, wipe down and install. Lube before riding the next day after everything has evaporated.

Life is too short to be anal about chain cleaning.