I didn’t know this sutta was that famous. 
But sure, since you ask so kindly, how can I refuse? 
(No promises for the future, though. I spend a bit too much time on this debate already, although it’s good to have all the main arguments of both sides in one place.)
I think this sutta actually also provides some little evidence for the disembodied jhanas. People always focus on one particular line, but overlook some other things. I’ll share my evidence at the end, first the line.
Venerable Analayo actually explains it in the same work I referred to, in footnotes on p85 and p123. Just like one can’t contemplate inside the jhanas, one also can not walk when in the jhanas, is what he says. I agree with Venerable Analayo on both accounts. Analayo’s footnotes are a bit terse, though, and I don’t know if he explained the grammatical details elsewhere. So here’s what’s going on in my opinion.
First the Buddha says he’s living a simple life in a forest, going for alms every day to a certain town, and that he’s attaining the jhanas. (It seems like he wasn’t teaching at those times.) Then there is a line which Bhikkhu Bodhi translates: “Then, brahmin, when I am in such a state (evaṃbhūto), if I walk back and forth, on that occasion my walking back and forth is celestial.”
Bhūto, which Venerable Bodhi translates as a present participle, is actually a past participle. But it is used broadly throughout the suttas, with varying meaning. Sometimes a present participle is an alright translation, but only when it has the sense of “being” in general, not “while I am in such a state” which I would say is too specific. It’s more like “while I am attaining such states in my life”.
So Bhante Sujato translates it as “when I’m practicing like this”, by which he means the Buddha is practicing those states in general, not that he’s actually in them while he’s walking. That seems to align with Monier-Williams dictionary which glosses evaṃbhūto as “of such a quality or nature”. In other words, when the Buddha is “of such a quality or nature” that he can easily attain the jhanas, then he walks “celestial”.
I think Sujato’s translation is better, also because the word “state” in Bodhi’s “when I am in such a state” is not in the Pali. Evaṃbhūto, literally “been such”, does not really refer to the jhanas. It refers to the Buddha himself being “such”. But what is that “such”? Is it only attaining the jhanas? Only the fourth which was mentioned just before? That would be the case if we take it all as a simple sequential sequence. Or does it include the whole passage, including him living in the forest, going for alms and such? I see no reason why it would only refer to the jhanas, because again, evaṃbhūto means “while I am being such”, rather than “while these states are such”.
At the end the Buddha says: “When I’m practicing like this, if I walk, at that time my walking is heavenly. […] if I stand, at that time my standing is heavenly. […] if I sit, at that time my sitting is heavenly. […] if I lie down, at that time my lying is heavenly.” To me in general the walking refers to going on almsround (which the sutta says he does before attaining jhanas), the standing is perhaps waiting for alms, the sitting is the meditation, and the lying down is his sleeping (and you can’t be in jhanas while sleeping, I think we all agree). I don’t really know why else he would lie down; apart from when he had aches in his later years, I don’t think he really ever did so for other reasons than sleeping.
Alternatively and perhaps more simply, bhūto may also be more literally translated as a past participle, meaning “when I have been such”, i.e. have been in the jhanas. A somewhat similar use of bhūto is found in SN46.30: “For in the past, venerable sir, when I was still a householder (agārika-bhūto), I did not have much concern for the Dhamma or the Sangha.” Here bhūto refers to a past that no longer is present. The PTS Dictionary also lists under bhūta: “pp. in predicative use […]: what has been or happened; viz mātu-bhūtā having been his mother.”
So, while there may be some ambiguity in evaṃbhūto, it’s certainly not a clear home-run for the embodied jhana view.
Now, the text provides some support for the disembodied view as well. People like to bring up the one line to argue one can attain jhanas while walking, but notice when the Buddha himself describes his practice, he doesn’t actually do that! He says: “Brahmin, when I am living supported by a village or town, I robe up in the morning and, taking my bowl and robe, enter the town or village for alms. After the meal, on my return from almsround, I enter within a forest. I gather up some grass or leaves into a pile and sit down cross-legged, setting my body straight, and establishing mindfulness in front of me.” Then, only after sitting down does he enter the jhanas. So we have to ask, if he could attain the jhanas while walking, why didn’t he do so on the way in or out of the village? 
Especially if that was so strongly indicated by evaṃbhūto in the same sutta? 
Of course, this is not a clear-cut evidence for the embodied view either. Because one can argue that perhaps he just didn’t feel like entering jhana while walking, or whatever. I wouldn’t be convinced by that particular argument. Maybe someone has a better one. But it is at least really interesting that in the very same sutta used to argue that one can attain jhana while walking, the Buddha himself actually says he sits down! 
And that’s not just here. There are a few other references where the Buddha either sits or lies down before attaining the jhanas, and none other where he’s walking. I belief it is quite telling as well. Some suggest that in jhana you can walk, do all sorts of things, supposedly even when you’re a puthujana. But then why would of all beings the Buddha apparently have to sit down? Again, why wasn’t he in jhana while he was walking in and out of town? Well, I have answers to this, but I’m asking it of others.
In the embodied view of jhanas vivicceva kāmehi is interpreted to mean something like being without sense desires, or being generally aloof from some unspecified “sensuality” or just not interested in sense objects. But while walking, the Buddha already was away from those things. So he had already fulfilled both the prerequisites for the first jhana: vivvicceva kāmehi and being without the hindrances (since enlightened beings have no more hindrances). But still he wasn’t in jhāna. Why not?
I think the phrase means to be fully (eva) separated (vivicca) from sense experiences (kāmehi), moving the mind away from the five senses. Then, when the Buddha was on almsround, he had only fulfilled one of the prerequisites, being without the hindrances. To fulfill the other he had to turn his mind inwards, away from the experiences of the five sense. To do that, he had to sit down. To me, that makes sense on many levels.
I hope all this also makes sense to some others, at least to the extent that they give this kind of meditation a serious try.
PS. It would also be difficult to walk while you’re not breathing. And in the fourth jhana the breath has ceased, say a handful of suttas.
Well, it clearly uses sukha in a different sense. The word does have different senses throughout the canon, so that’s not particularly strange. For example, parinibbana is also a type of sukha but of course there are no feelings there, so it is not sukha of the jhanas, which is a feeling. In the Uppaṭipāṭikasutta sukha must stand for piti, since that is what is abandoned in the third jhana. A bit weird, but the sutta seems to somewhat forcefully try to fit the four “faculties” (dukkha, sukha, domanassa, somanassa) onto the four jhanas, which, because these two sets of four don’t really match, in some cases is done in a somewhat creative way. Like, domanassa, which is said to be abandoned in the second jhana, seemingly refers to the vitakka vicara, which is a bit strange too. But this stretching and challenging of concepts is something that the Buddha did all the time.
Because whatever I say, you’ll likely relate it to the five senses again.
So please excuse me if this would be my last reply on the topic.