FWIW, priority waitlisting is a published benefit for elites.
https://www.delta.com/content/www/en...light-benefits
Footnote
#4 : Priority Waitlist Status: If the class of service preferred is not available when making a reservation, you'll receive priority waitlist status. This benefit is available only on a purchased ticket.
I don't know DL's policy, but some airlines expire the waitlist at T-24 hours, at which point I believe the fees for the GDS go up (but DL uses a proprietary system so could have flexibility). Regardless, it's not so straightforward, what happens is the waitlist will clear, and then your travel agent will have to issue the ticket. That can take a little bit of time, and I'm not sure if there's a blanket ticketing deadline, but certainly you'll need to be ticketed and checked in before that cutoff (usually 30-60 minutes by airport depending on baggage).
Also, some fares state in the rules that waitlisting is not permitted, but obviously the travel agent would know what fares can and can't be waitlisted.
As others alluded to, you can purchase a ticket for an available flight, and then try to standby at the airport for your desired flight, but there's really no guarantee there either. Also I'm not sure how DL handles those - AA, for example, can be very difficult and won't clear even a paid business class ticket directly into a no-show's seat, they'll first clear any upgrades for passengers
already confirmed on the flight, and then they'll offer you to fly coach if there's still room left.
Bottom line, you might want to consider a backup plan on another airline. Or at least confirm a reservation in coach if your schedule is that inflexible. And I would also consider an ExpertFlyer subscription and a flight alert for available inventory - in theory the waitlist process should grab anything that comes up, but you never know. If you get the EF alert, then you call the TA and make sure they grab the space.