I always check out in person at the front desk and 90% of the time they ask if they want a receipt emailed to my email on file or printed out (I'll always ask for the email, sometimes for the print out too). The other 10% of the time I'll have to ask them for an email copy. Regardless I've never had a problem doing it this way.
I always ask them to "force" the email and wait for it to show up on my phone. Sometimes the front desk assumes the automated process will email it later, and it doesn't.
The bigger issue is - why can't Marriott fix their damn website? Not sure if you've noticed, for the same stay, sometimes download bill works, other times it doesn't and other times it's not even an option. But you may need to wait a day or a week, and the option eventually returns to download the bill for that stay. My theory is that downloads are only broken when I'm trying to put together a last minute expense report.