I'm not sure, but ...
&air_date versus &first_air_date

&air_date = These are the exhibition and re-exhibition dates in any country in the world.
Old ones should appear because many have been re-displayed on new dates, more recent than the date you specified in the filter.

&first_air_date = It is the worldwide release date. Usually the one that appears in quotes, next to the Title.

So, I think you should use first_air_date.
To be more precise, use &first_air_date.gte=2019-06-16

Remembering that these dates may be the episode's airing.
If the series is long, with 15 seasons, 15 years of dates.

See here https://developers.themoviedb.org/3/discover/tv-discover

first_air_date seems to be a "fixed" value on an entire show and not for individual episodes. In other words, once the first episode has aired for a show, the first_air_date does not change (which is what I expect).

As far as I can tell, TMDb doesn't actually keep track of "re-exhibition" dates for individual episodes.

air_date tends to give results I expect, but only within the confines of a specific day or within the previous month. In my original example, it seems like a year-old is problematic.

As another example, assume the both the gte and lte were both set to 2019-06-16. This narrows the search to a single day.

Using the first_air_date variants yields only a handful of shows: all of which had their first episode that aired 2019-06-19. Using the air_date variants yields a number of shows, presumably ones that had an episode on 2019-06-19.

I'm interested in that latter scenario.

It seems that the rules are a little different from Movies.
For example "Breaking Bad".
I can only assume that there was an exhibition, of any episode, in any of the 41 countries, after 2019-06-16.
But that would need to be checked by looking at the 41 countries.
I think my suggestion only hindered the solution.
Excuse me.
I believe that Travis Bell can answer your question precisely.

By using air_date, you're filtering episode air dates. Just looking at your first example with Breaking Bad, there's a special that was added (which looks like it was intended to be El-Camino) in season 0.

Gotcha. That makes sense to me. Thanks for pointing out the specials, I forgot to check those.

I'd argue that that specific item probably doesn't belong as a special, but at least there's a reason.