July 4 was a Wednesday, but Garcia seemed like a vaguely patriotic guy (in a U.S. Blues sort of way!) who might want to gig on the 4th.
Now consider the following two scans:
According to a May 1973 ad in Bluegrass Unlimited, Old and in the Way and "Jerry Garcia (Grateful Dead)" were listed as scheduled to appear at the 1st Annual Pennsylvania Old Time Mountain-Country Gospel-Bluegrass & Blues Folk Music Arts Festival. I'll call that the 1APOTMCGBBFMAF for short. Based on fliers held in the archives of the Southern Folklife Collection Festival Files at the University of North Carolina, Chapel Hill (Folder #506), this seems to have been held in Valley View Park, York, PA ("off I-83").
Now, this scan is from the next month's issue of the same magazine. It no longer lists Garcia (Don Reno and Bill Harrell have been subbed in), but it does list Old and in the Way for the 4th. Vassar Clements is also billed for that day (with Doug Dillard), as he had been the month previous.
There are two fliers in the SFC collection, both undated. One lists "Jerry Garcia from Grateful Dead" (underneath "Peter Rowan from Seatrain" -- think they were trying to appeal to the youth/rock audience?). The other does not, and makes no mention of OAITW, either.
Someone who was involved in the bluegrass festival scene at this time says that he doesn't remember seeing Garcia at this 4th of July festival. Combined with the printed materials, this seems to suggest that OAITW/Jerry were originally scheduled to play, or at least discussions were happening, but it didn't come to fruition.
This still leaves open the question of what Jerry was doing on Independence Day 1973, but I do doubt that he attended the 1APOTMCGBBFMAF.
The summer of 1973 bluegrass festival scene certainly warrants a post on its own.