What is Reddit's opinion of Plywood Circles 4 inch, 1/8 Inch Thick, Pack of 25 Unfinished Wood Plywood Circles for Crafts with Rustic Burnt Edges, by Woodpeckers?

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Hole saws usually use a centering bit. Which would put a hole in the middle...

It may be easier to buy or make a jig to use with a band or table saw. But that may he more expensive than you want.

I'll bet that a store like Michael's or a similar craft shop would sell rounds of plywood for relatively cheap. Or they are probably even available on Amazon.

https://www.amazon.com/Woodpeckers-Unfinished-Creating-Christmas-Decorated/dp/B01MECLB9Y/ref=mp_s_a_1_6?ie=UTF8&qid=1524768991&sr=8-6&pi=AC_SX236_SY340_QL65&keywords=plywood+round&dpPl=1&dpID=41P1UtUJk2L&ref=plSrch

(I was limited in my search to what ships to Canada)

From there you just need a method to attach the wood to the peg to the wall... and I'll bet you could get away with black iron pipe and pipe fittings which would attach to the plywood and the wall quite easily with screws.

https://www.homedepot.com/b/Plumbing-Pipes-Fittings-Black-Pipe-Fittings/N-5yc1vZbqn5

Flanges like this and a small iron pipe section would allow you to use small.screws to attach to the plywood and heavy duty ones to the wall in a stud or drywall anchor. This is WAY overkill to hang up a jacket, but would be very simple to do with adjustable depth because you can get the pipe cut and threaded at home depot to your required length.

Stain it and seal it and Bob's your uncle.