I just want to note that there is similarities in those two lines to other minstrelsy numbers in terms of context: Oh My Darling Clementine. It is a simple address that fits in the context of the opera as a whole but alone it is rather plain.
The second note is the translation gets rid of several poetic features. Namely the repetition on the second line (è bello bello) as well as the alliteration that begins each line (O mio. . ., mi. . .) and the alliteration on the two strong words (babbino, bello bello).
Compare it to the similar address in the refrain of the song I cite:
Oh my darling
Oh my darling
Oh my darling, Clementine
We have a repetition of the first line, as well as a rhyme. The rhythm is trochaic save for the last word which is interrupted to conclude the hypermeasure with a cut syllable. This is done over music that escalates in tension and release over a single phrase.
In O mio babbino caro, the refrain descends in a lullaby fashion that in the context of the whole opera is fleshed out further. Why is she saying "Oh dear father!" so? One needs to know the rest of the opera.
But as Chris says, the fact that you don't know what the lyrics are saying but can hear the music and understand that what the song is supposed to be sweet and beautiful, it is not surprising that you expect the lyrics to be more intricate or more self-contained in gravity. This problem, if you want to call it that, is because you are importing context onto the music on which you do not know the context of. It just sounds good to your ear.