I agree with bertsmobile1. Not to be negative or to say it can't be done. There are so many things to take into account. Drilling and tapping tool hardened steel is a feet in itself. It can be done but the crankshaft would need to be removed from the engine. It would take a access to a milling machine / Bridgeport / something .... will need a ton of coolant and more than an average drill bit. Let's say for the sake of argument that you are able to get some sort of hole in the end with a drill, and even tap that hole, if it is not dead on center all sorts of problems will arise. Engine vibration, etc. etc. Most concerning would be the chance of compromising the integrity of the metal while drilling the hole. The experience of a crankshaft exploding at 3,600 rpm is like no other! The other things to look at are the crank diameter? Also, the length of the crankshaft (PTO) extending through the bottom. If the length is not the same you can spend a countless amount of time trying to find a blade adapter that will fit, and make up the remaining balance of length. Too short and it doesn't cut or discharge correctly. Too long and it may hit the ground.
If you are dead set on trying to drill it, I would offer this advice: Buy a short length of 5/8 or even 3/4 stock rod. Most hardware stores will sell it. It is not hardened so it will be significantly easier to "mess" with. Run a trial on it. Try to put an inch or 1.5 inch hole directly in the center and tap it. Then get some measuring tools and see how it comes out. That way if you find it next to impossible, then you are only out a few bucks in metal. I have ruined a few completely good engines trying to "fabricate" and in my experience it only adds insult to injury.
I don't know where you are located but in my area, mower engines are a dime a dozen. In all things, the worst time to try to find/buy is when you need it. But used mower engines should be around and relatively inexpensive. I'll bet the second you aren't in a hurry to find one, they will manifest out of thin air. (at least it seems to work that way for me)
Lastly, what kind of mower are you working on? I know you said a Briggs engine, but what brand is the mower?
This is just my shared experience. Take it or leave it. Whatever the choice, I hope it works out for you.