Well, we've got the Hawaiian humuhumunukunukuapua'a (which is common enough to be the state fish of Hawaii) and lauwiliwilinukunuku'oi'oi (which is unspecific enough as to refer to two different species, tho they are admittedly almost identical so I don't see why any language except science
should differentiate between them), after all.
But they are of course distinct species and not entire families of animals.
You can see some pretty nice oligosynthesis-like formations in some languages deciding on neologisms for new concepts, tho. Cree for 'horse' is supposedly literally "big dog" (and as a side note,
this Navaho etymology for dog is kind of interesting), or Navaho
chidí naaʼnaʼí beeʼeldǫǫhtsoh bikááʼ dah naaznilígíí for the English monosyllable 'tank'. I don't think it's unreasonable that some things that seem basic and short in one language should be much longer in others.
This happens sometimes when the language lacks the concept natively and opts for a neologism instead of a loanword, and I suppose one could say that Vyrmag is meant to have, well, almost nothing natively.
As a living language, tho, which OP wants it to be, I would expect short forms of common words to crop up eventually.
The thing is that sometimes you just don't have to be specific. Some languages are ridiculously specific IMO. I don't see the point of distinguishing jam and marmalade. Distinguishing certain specific species of animals isn't always relevant and can be left ambiguous if the context allows for it. The lengthier forms would only be used when completely necessary for disambiguation, which is a reasonable enough situation for one to afford a few extra syllables.
That said, we all know the problems with true oligosynthesis, and I'm not denying those. I just think people often point out a lot of problems that aren't really problems due to native bias.
And some languages
do have words that cover multiple entirely different types of animals based solely on shared features like 'armoured', and don't use specifics unless they have to. Many languages can use 'bird' for anything flying, including for example insects (Danish sommerfugl, literally "summerbird", meaning 'butterfly) or aeroplane (aviation; avion), or 'fish' for anything swimming (English 'jellyfish'; Swedish bläckfisk, literally "inkfish", meaning 'squid') and so on. And it works out just fine.
And it's very common. I feel often Anglophones are so accustomed to having obscure loanwords for every specific thing that they don't realise that most other languages simply use native, oft ambiguous, compounds or derivations like these, and often make fewer colloquial distinctions than English.
Just the other day I talked to someone who found the Faroese word for rhinoceros, meaning literally the same thing as the Greek word but without being obscure to a Germanic speaker, and being similarly formed in most Germanic languages besides English, ridiculous because it was so literal – but even rhinoceros itself is literal if you know Greek. And very ambiguous. Nosehorn could be anything. It could be an iguana.
On the opposite end, languages have many words that are ambiguous unless you know that they have specific usage, including even English 'shellfish' or the even more ambiguous Swedish translation, skaldjur, literally just "shell animal". Skalbagge means 'beetle' but bagge alone means 'ram'.
