MN 137 - Directed only to Stream Entrants (and beyond)?

You will not have joy, but aside of tee-break stories of sectarians and householder it is the joy opposite householder-joy/satisfaction:

"And what are the six kinds of household joy [in MN137] ? The joy that arises when one regards as an acquisition the acquisition of idea cognizable by the intelect — agreeable, pleasing, charming, endearing, connected with worldly baits — or when one recalls the previous acquisition of such forms after they have passed, ceased, & changed: That is called household joy. (Similarly with form, sounds, smells, tastes, tactile & sensations).

It's archived when leaving home, having gained higher right view, and steady when gained non-return.

And once gained one does no more take what is not given and provides what is the deal to make use of something for ones gain.

The Sutta is given for distinction, that one knows the different intellectually and put it into prove, and certain explainings are given by the translator as well.

"'With regard to them, depending on this, abandon that': thus was it said. And in reference to what was it said?

As in regard of "To whom was the Sutta directed?": Those who have left home/house.

[Note: this is a gift of Dhamma, not thought for trade, stakes, exchanges or other gains subject toward decay and should be deleted if it's not giften to give in Dhammic conditions]

Is this achieved by intellectual pondering about the six senses (and meditative insight into the three characteristics) or just pondering alone?

I'd say the former, but then this Sutta is primarily directed towards Stream Entrants, no?

Intellectual pondering and/or meditative insight probably won't cut it. While there's no mentioning of the Sutta being directed only towards Stream Entrants, the Comy.'s explanation seems to suggest it does require one who has already established insight-knowledge:

Note. 1: [Six kinds of joy] “Based on the household life” means connected with the cords of sensual pleasure; [Six kinds of joy] “based on renunciation” means connected with insight.

Note. 2: "Herein, what are the six kinds of joy based on renunciation? When, by knowing the impermanence, change, fading away, and cessation of forms, one sees as it actually is with proper wisdom that forms both formerly and now are all impermanent, suffering, and subject to change, joy arises. Such joy as this is called joy based on renunciation." And explanation: "This is joy that arises when one has established insight and is sitting observing the breaking-up of formations with a flow of sharp and bright insight-knowledge focussed on the formations"

The way my teacher explained a crucial point about this was summarized with a single but powerful word: Immediacy!

At some point in our practice our familiarity with Dharma should go beyond it being something "over there" that we study and try to understand, and become something very personal that is happening "right here" in our own immediate experience, second-by-second.

This type of joy that arises in case of mature student, is part of this breakthrough to the Immediacy of Dharma. On the initial phases of the breakthrough, we keep discovering more and more real life microsituations that turn out to be perfectly described by some ancient scripture, and this gives a sense of wonder and joy. And then as we get used to this match, we get a kind of relaxed bliss of a traveler who reached the oasis, had enough food and water to recover, and is now resting in the shade (to give a traditional image of what "bliss" means in this context).

Specifically as it pertains to Transience/Inconstancy of all configurations, once you clearly see that every frustration you used to take seriously is a temporary arrangement of clouds that actually shifts beyond pain fairly quickly, as long as you don't hold on to it, -- this immediate observation gives a sort of deep peace and a sense of relief that you experience continuously as you interact with all kinds of situations.

So to answer your question explicitly, it's not just intellectual pondering nor is it meditative insight, as it is the immediate observation (directly seeing) it all in your own life.