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Unless you ride in very wet conditions (in which case your freewheel should be a good sealed or shielded one like Dura-Ace, Sante or 600), there in no need for heavy lubrication inside.
The most I ever do for a freewheel is to blast it out with degreaser and then water (if it is completely unsealed), and then dry and re-oil it, and "whirl" out the excess oil outside using my arm rotation. Or I can lay it on a rag on it's end and come back the next day.

Freewheel bearings won't wear out from some grit in there. The worst thing that happens to freewheels is when they get internal rust and then they are goners that are only good for salvaging cogs from.

I don't clean chains, other than to wipe down thoroughly with terry cloth after lubing with a diluted ("watery") style of chain lube.
I used to apply degreaser to chains from the left side after leaning the bike toward the right, so as to keep any dripping away from the critical bottom bracket and rear axle bearings, but since I don't rescue crapped-out chains any more this all is in the past.

Hopefully, the good, cheap chains that we have been blessed with don't come into short supply like many other bike-related bits of late.

Increasingly I am cautious not to empty out spray cans or end up having to pour away contaminated solvents. I use mechanical means to strip off stubborn deposits and then just the rag (maybe with a few drops of Finish Line Citrus Degreaser). The spray cleaners are expensive, and just blasting away is hugely wasteful plus environmentally unfriendly to the extreme. I recommend brushes and various metal scrapers (like a 6" machinist's rule or a screwdriver) to remove chain-path detritus before a final wipe-down using a rag or shop towel.

Chains that are lubed and wiped periodically should see absolutely no buildup of grunge over time, as long as a thinned (solvent-diluted) type of chain lube is used.