1. As you've already mentioned, if you were to stay in the same area rent would be cheaper than a mortgage. Rent is always cheaper than a mortgage for equivalent real estate in the same area. So even if you moved, there would still be really strong reasons for renting since it would still be cheaper than buying a place. So your argument of $2400 rent vs $2400 mortgage is just wrong since moving to a different area will result in $1200 rent (or less).
2. Unlike a mortgage, you're not going to be spending anywhere close to $864k upfront. At most it's only going to be $7200 up front. If you were going to buy a place in the same neighborhood, it's probably going to be about $400,000 upfront, not including closing costs and other fees. Not including stuff like renovation or housing insurance either. Don't forget HOA for certain areas as well.
So if you have 500k in cash, if you rented you've about $492,800 for investing or for starting something up. If you bought a place in the same area, you have nothing for investing or starting something. So this "$0k vs Whatever the house is worth" argument is really flawed, especially when you consider that your capital is "stuck" with a huge opportunity cost unless 1) it appreciates a lot and you're willing to BOTH sell and move into a cheaper place that isn't as nice in terms of location or features (if you're not you're just stuck living in a nice expensive place which isn't a bad thing but remember opportunity cost), or 2) you're a professional house flipper (your home really is an investment and not just a place to live), and you repeat this process every 1-4 years.
If you haven't already guessed, I'm not a renter and I just figured that you want something more than normal people since you roam HN.