What's a positive word to describe happily stumbling around on a pleasant afternoon, accomplishing nothing in particular?

How about "amble"?

https://www.merriam-webster.com/dictionary/amble

"Bilbo was enjoying his mead, ambling around the yard, not accomplishing anything in particular this summer afternoon."

The origin of "amble" is from a Latin word meaing "walk about", so it would primarily denote a walk. It doesn't depict any parallel activity undertaken while walking.

As to the style and intent of the walk: to walk about suggests that the walk is not undertaken specifically to get from one place to another: "about" in this context feels like walking "to and fro", or perhaps "in a random way".

https://www.vocabulary.com/dictionary/amble

Amble comes from the Latin ambulare, which means "to walk about," as in ambulatory.

Lastly, there is - to me personally - an intangible positive, relaxed quality to the word "amble"; just something about the way it sounds.

That is called pottering (around).

Potter (around): To move around without hurrying, and in a relaxed and pleasant way.

Example: I spent the afternoon pottering around the garden doing a few odd jobs. [Cambridge English dictionary]

In your sentence, you could say:

Bilbo was enjoying his mead, pottering around the yard, not accomplishing anything in particular this summer afternoon

pootle, v.

OED British colloquial. intransitive. To move or travel in a leisurely manner; [...] Frequently with around, along, about.

1988 Bicycle Midsummer 57/1 Pootling through Nottingham recently, I came across a branch of a shop called Concept Man.

2020 http://more-to.org/about/ And I love walking, enjoying several expeditions in the greater ranges of the Himalayas and the Andes, as well passionate ramblings around the mountains of Wales. When not walking, you’ll often find me pootling about in the garden, or sitting out with friends and a nice bottle of wine…

No-one so far has mentioned bimble (which is British English):

Bimble at Urban Dictionary:

To amble without real aim, yet in a friendly and harmless manner. It's not required to acheive nothing, though it is a frequent side effect. Bimbling can be made a little more business like with a slight hunch of the shoulders. "Tron and Enid whiled away many a Sunday afternoon on a pleasant bimble round the shops."

Bimble at Wiktionary:

Bimble (noun), (plural bimbles), (chiefly Britain): A gentle, meandering walk with no particular haste or purpose.

Bimble (verb), (third-person singular simple present bimbles, present participle bimbling, simple past and past participle bimbled), (chiefly Britain, intransitive): To walk with no particular haste or purpose.

"Bilbo was enjoying his mead, bimbling around the yard, not accomplishing anything in particular this summer afternoon."

Only thing that comes to mind is "Saunter," which is a leisurely walk, stroll etcetera.

"Bilbo was enjoying his mead, sauntering around the yard..."