Most of my in-person urgent travel appointments at US passport agency offices in the country have been done for US passports that were still valid for travel at the time of in-person application. Sometimes they look at the application travel date info or ask when the imminent travel is so they can figure out how fast they have to move the file, but don’t worry about that being a problem.

If you have an appointment, they shouldn’t care whether your current US passport is still valid for travel or not as long as it’s still useful as proof of citizenship/identity for the paid application — assuming you are using it for that proof for passport eligibility.

If at the time of paying and submitting your properly completed application they say they can’t turn it fast enough for you to make the July 5th evening flight and make an issue of it, just note that the airline may be more willing to freely rebook you on standby within 24 hours of originally scheduled travel and early enough next day pick up is better than nothing.