Chord inversion is effectively helping you to build a voicing / melody e.g. in the upper notes.
So, not inverting a chord could be considered ineffective from a melodic point of view.
Regarding the bass progression, you can still play the root with the left hand, if needed.
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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.
music.stackexchange.com/users/18896/topo Reinstate Monica
You don't lose the harmonic function of the chord.
So, if you play a V in root position, 1st, 2nd or whichever inversion, it will work the same, no matter which note is on the bass.
This is
because you don't change the chord per se, you just change the way the notes are arranged ( and especially the bass note )
Inversions are an easy way to make a simple chord progression interesting.
It'd be boring to play the same chords over and over again, so you can create more interesting patterns by inverting them.
Also, the bass lines become way more interesting if you use inverted chords.
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It's fine.
Though there are certainly cases where you'd be more confident or not.
For instance if playing with a group that includes a bass player or bass-like instrument that is playing a bass line, it definitely doesn't matter if you play it.
On the other hand if it's a song where the bass line is not just supporting the harmony but is a distinct and important riff itself, then somebody should probably be playing it.
So when playing such a piece solo it may even be best to devote your left hand to it entirely.
But generally the answer is that it's fine.
The harmonic function will still be felt.
In fact, if you don't play inversions—or more accurately if your voicings aren't voice led from one to the next—you may be missing the some of the point of the harmony in that instead of a note moving to the next in a smooth and pleasing manner, it may make a large and potentially awkward intervalic jump.
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Yes, the defining bass note (root) does get lost a little.
No,the bass progression doesn't get lost.
It's hardly going to be a progression if only the root notes are played in the bass.
Other passing notes are usually involved, and they don't always lead directly to the next root.
Yes, they can, but that becomes quite predictable, and in music, it's good to do things that the listener isn't expecting.
So, playing, say, a C under a C chord is fine - especially in C, but with C,C,B,Bb in that bar, there's a good reason to play an A in the next bar, which looks like it's going to be an F chord.
On a G abar, going back to C, a B bass works well, followed by an almost inevitable root C.
It all works, and really, there's no 'problem' to get around.
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There is a difference between the root progression of chords and the actual bass notes used.
The OP has touched upon what happens.
One may have (for example) a chord progression of C-B0-C with the bass being E-D-C, it's not uncommon.
Note that the first two chords are in first inversion.
One hears a C chord then a B0 (diminished)
chord then another C chord but simultaneously hears a stepwise descending bass line.
First inversions do not radically change the sound of a chord (they do change it though).
First inversions are useful to keep the bass line moving when bass line with jumps may not be suitable.
That's a decision a composer or arranger must make.
Another possibility would be a chords C-B-C with chords being C-G-C or C-G6-C; in the first case the bass is C-G-C which is fine; in the second case the bass line would be C-B-C which moves less; also fine, but different.
Second inversion chords change the sound a bit more and are thus more complex to analyze.
Sometimes the second inversions are used in a manner that the do not sound like the main chord at all, but rather as another chord with non-chord notes attached.
In the classical Dm-C64-G7-C, the second inversion C64 sounds a bit like a G chord with a couple of notes moving to their "correct" places later.
In other places a C-G64-C6, the G sounds like a G chord with a stepwise moving bass.
Much depends on the point of view of the composer (or analyst); one can thing of a succession of chords or a bunch of melodies which coincide to produce chords at different places.
Both are useful.
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If we invert the chords on a simple song on the piano, then we lose the tonic in the bass.  So doesn't the bass progression get lost (?)  I heard somewhere that to  listen by ear one has to listen to the bass notes . But if we change the bass around then how would one hear the chord? doesn't the chord get muddied or something?  However it seems to me that most piano players invert their chords. So how do people get around this "problem"?
If you invert a chord, doesn't it lose its defining bass note?
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Chord inversion is effectively helping you to build a voicing / melody e.g. in the upper notes. Not inverting a chord could be considered ineffective from a melodic point of view.  Regarding the bass progression, you can still play the root with the left hand, if needed. Chord inversion is effectively helping you to build a voicing / melody e.g. in the upper notes. 
Chord inversion is effectively helping you to build a voicing / melody e.g. in the upper notes. Not inverting a chord could be considered ineffective from a melodic point of view.