I believe the answer to your question is addressed on the Delta site.
There does seem to be a unique requirement, applicable to day passes, that travel be on Delta (or Connection carrier) as follows:

One-visit Pass
The cost of a one-visit pass may be credited within 30 days of purchase toward the cost of a one-year membership.
No guests are permitted with a one-visit pass.
A one-visit pass may be used only in conjunction with same-day ticketed air travel on Delta or Delta Connection® carriers.