The way one of my music teachers explained it to me was that a root position triad was like a chair with four legs - very stable; a first inversion was like a chair with three legs - still stable, but less so; and a second inversion was like a chair with two legs - rather unstable!
It's not really 'accurate', of course, just a fun way of explaining it. And yet he (and I) would agree that yes - you do sometimes 'lose' something when you invert a chord, whether you think it's some sense of 'stability', or (as I do) some of the sense of consonance.
However, this is a good thing! Why? because we don't necessarily want every moment in our music to feel completely concordant, settled, and resolved; we want to have moments of lower consonance and less stability that lead the listener's ear to expect the piece to move forward towards consonance and stability.
The other benefit it gives you is to allow individual musical lines that move in a more interesting way. The shape of the bass line can capture the ear like any other melody, and by allowing it to move away from the root of the chord to other chord tones, you can give it a lot more freedom.