In Defense of Latin

It means that every conjugation of an active verb has a corresponding passive form, as in your link: esąs - buvęs - būsįas versus esamas - būtas - būsimas . But that's underselling it a bit, because Latin actually had a full 3-dimensional matrix of conjugations: time-based tense (past, present, future), goal-based tense (completed or ongoing), and action (active or passive), with accompanying participles for each case. To compare with your link, I notice Lithuanian seems to lack passive participles ("while being beaten" is the corresponding example), while keeping some of the circumstantial conjugations (frequency, intent, necessity, conjunct phrases) that older languages did have, but Latin did away with.

In more analytical languages, most of these constructs have been replaced with auxiliary verbs (as in English: I see versus I'm seen, I'm watching versus I'm being watched). With full conjugations, entire sentences can change in tense or action without affecting word order. This may not seem that special if your native language is like that, but as the OP started out saying, it allows for much more expressiveness in (predominantly written) language.

Lithuanian is very conservative in some aspects, and very progressive in others. For example, it has lost the neutral gender and it has innovated some cases that didn't exist in PIE. There is no scientific, consistent way to call it as a whole more archaic than any other language.

If I got it right, Lithuanian does have passive participle (neveikiamasis dalyvis). In the case of "while being eaten" it would be "valgomas". "While being beaten" -> "mušamas". He was silent while being beaten -> Mušamas jis tylėjo.

To quote Wikipedia:

    the Lithuanian language is unique for having fourteen different
    participial forms of the verb, which can be grouped into five
    when accounting for inflection by tense. Some of these are also
    inflected by gender and case. For example, the verb eiti ("to
    go, to walk") has the active participle forms einąs/einantis
    ("going, walking", present tense), ėjęs (past tense), eisiąs
    (future tense), eidavęs (past frequentative tense), the passive 
    participle forms einamas ("being walked", present tense), eitas 
    (“walked” past tense), eisimas (future tense), the adverbial 
    participles einant ("while [he, different subject] is walking" 
    present tense), ėjus (past tense), eisiant (future tense), 
    eidavus (past frequentative tense), the semi-participle eidamas 
    ("while [he, the same subject] is going, walking") and the
    participle of necessity eitinas ("that which needs to be 
    walked"). The active, passive, and the semi-participles are 
    inflected by gender, and the active, passive, and necessity 
    ones are inflected by case.

I've studied Latin in school (and plan to do it again in the future, I'd like to be able to read it semi-fluently eventually). I really enjoy the language but I'd be extremely wary of using qualifiers like "powerful" for languages. That generally leads us towards extremely bad linguistics. Being more synthetic doesn't mean that you're more powerful per-se. You haven't said that exactly of course but I encounter this sentiment a lot in English-speaking circles, with people saying things like English is "dumbed down" or simpler because it prefers analytic constructs to synthetic ones. I also remember a Brazilian fellow saying that Brazilian Portuguese was worsening because people favored the analytic future "vou comer" over the synthetic "comerei" (which is amusing because this synthetic future comes from an analytic construct "comer hei", it's not the Latin future tense).

As a sibling comment points out, these features exist in may other languages, for instance Slavic languages such as Russian or Polish. Besides, even though this flexibility is theoretically achievable, in practice the day-to-day language is filled with more calcified idioms and ways of speech. You could, in English, say "to you this book I give" instead of "I give this book to you" but I'm sure most agree that's rather slanted even though I believe this is perfectly grammatical. A poet could write the former, but if you say something like that in colloquial speech people are going to wonder if you're having a stroke.

Judging the "power" and flexibility of Latin through the verses of Ovid is like judging English through Milton's Paradise Lost. Only educated speakers would come even close to even understand properly that level of language, let alone construct it.

Of course that's just speculation on my part but given that modern Romance lost the case system almost entirely it seems like there wasn't a lot of friction here. If common Vulgar Latin speakers really valued word shuffling surely they'd have found a way to maintain the cases I'm sure. I could also point out that in Russian (a language I've been studying a lot lately) it's very easy to build a sentence that seems grammatically right but sounds off to a native speaker. For instance Russian speakers tend to prefer to put locative expressions at the beginning on the sentence, so they say "on the table is the fork" instead of "the fork is on the table" which may sound slanted or putting emphasis on a different part of the sentence.

All of those, which can be applied to prefixed verbs [1], such as the ones below derived from the verb "valgyti" (to eat) [2]:

antsivalgyti - (with genitive) to eat one's fill (of); to fill up (on)

apvalgyti - (with accusative) to eat (someone) out of house and home

įvalgyti - to be able to eat up; to be able to consume

išvalgyti - to eat up; to devour; to eat away

pravalgyti - to spend (one's money) on food

privalgyti - to eat one's fill

suvalgyti - to eat up; to consume

užvalgyti - to eat a snack

[1]: http://www.lituanus.org/1991_4/91_4_06.htm

[2]: https://en.wiktionary.org/wiki/valgyti#Derived_terms