My favorite podcast is probably Writing Excuses, with the tagline "Fifteen minutes long, because you're in a hurry, and we're not that smart." They manage to signal the entire tone, plus an idea what to expect in the re-usable tagline, which lasts about 5 seconds. There's a little more to how they structure their opening, but it is amazingly concise and consistent – so that's probably lesson #1. If you have a repeating intro, make it as short and as comprehensive to the overall series as possible.
They also squeeze in the day's episode title, but I don't think it really works because the catchphrase is the thing that you come to expect, and the individual titles (spoken within the introduction) don't really register or linger in your mind. Listeners are not waiting to learn the title and subject with their ears, this information is in the metadata of the podcast itself, so lesson #2: Don't bother individualizing the intro for informational purposes whether that is a title or an announcement, it is like opening credits that are unnecessary because this "data" is not searchable or indexable by library systems. Put individual episode data in the file's metadata, not the episode itself.
Writing Excuses then do 1 quick joke (probably an excerpt or outtake, possibly staged for the format because it rarely appears in the show) that has nothing to do with the podcast, but is maybe a joke on the title or subject – timely, but not relevant. Longer (1hr) podcasts use this as a teaser and pull a funny moment from the longer show, but in a concise show that's only a few minutes this should probably be a non-sequitur or a foil to the actual topic. It helps "cleanse the pallet" by way of a "hot take" which will soon be debunked or expanded. It is random and off-topic probably, so putting it up-front is the only way to get it in before the "proper" info. You also don't want to end with a foil, since the ending should be the strongest conclusion (the "least" hot take) that you want the listener to remember.
After that they cut to the topic and stay there (except for a segment when they feature one writer's current project as a "commercial", which sometimes fits the theme but not always).
I think the biggest issue is not really how long each episode lasts (despite the tagline, their podcasts are not actually 15min), but how much time is intended to pass between the end of one podcast and the start of the next. For Writing Excuses, they include "homework" that you are suppose to do between podcasts, and the topic of the next show doesn't necessarily build on ideas from the one before. Shows can be listened to out-of-order. Of course you can binge, but it isn't inherent to binging, they have a regular release schedule but the season is not sequential or dependent on other podcasts during that season.
Shows that do build on previous concepts, or are intended to be binged one-after-another, probably need to lose the episodic introduction entirely. Instead they should have one segment that explains the "group" (however the topic is organized) and the individual episodes are designed to flow together more like book chapters. The timing of the episodes' release is not important, since they are all intended to flow sequentially to create the larger topic. I would still add the "hot take" or foil at the beginning of the individual episodes, the pallet cleansing effect will really help here.
Credits, music, standard intros and outros will all become obnoxious on binging, but they might feel more essential to a professional show that is only heard once-in-a-while, or when the topics do not flow together sequentially. The shorter the show, the more often the listener will hear the repetitive parts that play every time.
This is just my "hot take", but thanks for raising the topic!