Is the age of a checking account worth anything?

Many checks do not include the date that the account was opened. To the extent that checks do include such a date, this would be just to add another personal touch, like AAA does with membership cards. From the bank's point of view, checks are just instructions. Extras like an address and phone number are just there to reassure the person accepting the check as payment. Sophisticated retailers such as Walmart require a photo ID plus third-party verification (TeleCheck, etc.). I don't know that any retailers would give the date that the account was opened any additional weight.

having a long-opened checking account represented a sort of stability or "credit-worthiness" when presenting a check to a merchant.

Think rather "stability". The problem is that even then you could order checks with a high starting number.

She spoke of it as a badge of honor and mentioned how important it was to keep a long history of keeping a checking account active.

I felt the same too. Had the same checking account for 40 years, which got automatically closed during divorce due to disused (because I'd added her as co-owner). Was very displeased at myself for not keeping it a bit active.

But you know what?? My credit score didn't change a bit, and companies have kept accepting the few checks that I do write.

So, to answer your questions:

Is the age of a checking account worth anything?

Only to you. (Which can be important, but only to you.)

why is the date that I opened my checking account still printed on my checks?

Either because that's the option you chose when you first ordered checks, or that's the default which the bank has decided on.

I do recall when I worked retail that we didn't accept so-called "starter checks" which I think by our definition meant checks where the number was below 100, or checks that were not personalized (didn't have your name and address pre-printed because I guess the bank gives them to you to use right away before your printed ones arrive in the mail?), or both.

Hard to remember the specifics now, and given the way retail tends to operate stuff like that can be at the store level or even depend on which manager is working, so I'm not certain it was really based on sound principles in the first place.

Beyond that, we had no clue how old an account was and didn't think about it.

Editing in more detail as I remember it: we also required ID, and wrote the state drivers license or ID number on the check.

So when you think about it, the requirement was likely more about identification and being able to track down people who passed bad checks (their pre-printed name and address, matching their state ID), than it was strictly about account age.