Based on my very limited experience and the posts above, I'd say sign languages don't really have the luxury of having many "synonyms" with slightly different connotations, at least in the same way you'd list words in a dictionary. jbergstroem mentions "always trying to find the simplest way of explaining yourself", so I think the idea is people just say things how they are, in simple terms. I imagine ideas like politeness are instead conveyed in other body language, facial expressions maybe, instead of dancing with words.

But I'm just speculating, I'd love to hear someone's firsthand experiences here.

Limited practical knowledge here (beginner ASL and some time in a linguistics research lab) - there are fewer synonyms, but facial expression and other movements make a huge difference. Much like German adds suffixes that can really change the meanings of words, a eyebrow or tongue movement can modify a sign in ASL.

Another reason ASL has fewer synonyms is that it’s not a horrible amalgamation of a ton of different languages like English is (English has a weirdly large vocabulary, like 3 times as large as some other languages). ASL is a direct descendant of French SL, no weird mixing. Moroccan SL, in contrast, is a descendent/hybrid of several languages and it, as an example, has like a million different handshapes for different kinds of insects.

Further, a word-sentence model is too limited to describe sign languages (I know this applies ASL and Moroccan SL, not sure about others). For example, a large set of concepts are expressed through classifiers. Classifiers are systems of hand-shapes, spatial locations (e.g. side of head vs in front of chest), and motions which can specify a objects (including its shape, size, and location) and also the objects’s verb and relationship with other objects. Some people mistakenly think classifiers are a form of pantomime, but classifier motions and hand shapes have specified and limited meanings (e.g. the flat rectangular object classifier is not making a rectangle shape, it’s more of a pinching motion). That’s just one example of how sign languages diverge from a word-sentence model, many more differences exist.

The main take away is that sign languages work differently from spoken languages, it’s not that easy to compare 1-to-1. There’s definitely no word that means “rotund” in ASL, but there are certainly grammar structures that let you express that.

Although you could, there’s no need to soften or euphemize physical characteristics in American Deaf culture. You can just sign “that fat person over there” and it’s not considered rude.

I don't think so.

My deaf BSL instructor said that it was about being understood clearly, and just about deaf culture. If you're in a room of people signing everyone can see you're saying "who is the fat guy at the bar" so there's no point in being euphemistic. Might be a UK deaf culture thing.

Any time you get enough deaf people in one place, who don't already have a signed language, they invent one. In times and places where signed languages have been suppressed, they've popped up all over the place at schools for the deaf. There's not much reason for any of the languages to share grammar and such with each other, since they're effectively unrelated. You wouldn't expect Japanese and Navaho to use similar grammar or vocabulary, right? Same here.

[Not Deaf, but I studied ASL and deaf culture for a few years at 2 colleges and in private lessons.]

There certainly are relationships between sign languages; A(merican)SL and French SL are arguably two dialects of the same language, or at least they were in relatively recent memory. Others are distantly related or unrelated; B(ritish)SL is absolutely unintelligible to signers of ASL and arose separately, for instance. Many of the modern global sign languages are related to one or the other of those families (and might be "dialects" as you want to call them, or might be further apart and thus merely "related languages"), but there are a few others that are not, including one—Nicaraguan Sign Language—that arose from scratch less than 40 years ago.