They are all fairly well standard
7/8" push mowers & walk behinds
1" domestic ride ons
1.125" commercial ride ons & construction equipment
Forget about CL or Evilbay
Get yourself a brand new engine from one of the following
The Surplus Centre
Small Engine Supplies
Small Engine Warehouse
The latter 2 have an app which sort of works if it actually recognises your mower model number or the original engine number
Your mower has a 1" crank 3 & 5/32 " long tapped 7/16 as will almost all of the engines.
My pick would be this one
https://www.smallenginesuppliers.com/shop/html/pages/products/briggs_vertical_shaft_engines4353.html
Having the pull start is a big plus as you can start the mower with a flat battery.
If your mower has a tank behind the dash then all is good, if it is under the seat you will need the fuel pump & the dip stick tube with a take off for the pump.
They all spin at the same speed so a bigger engine will not go any faster than a smaller engine.
If you cut irregularly or do rough cutting in a paddock then a bigger engine might be worth the extra.
Personally I tell customers to ignore the Hp and just look at the series .
Any 31oooo engine should be the exact same dimensions so will drop right in , right up to bolting onto the muffler.
If ou don't have a muffler then buy an engine that has one as mufflers are part of the mower, not the engine so usually run to a 3 figure number so a couple of minutes with some tin snips can cave you $ 100.