I understand the functionality of rootless chords in a band setting where there is a bass player, but rootless chords are supposedly used in solo piano as well (the Jazz Piano Book says this, and I think Bill Evans is a prime example).
However in solo piano what is the point of rootless chords? If a play rootless D-7, then I play F-A-C-E which is the same as Fmaj7, so why not notate the chord as Fmaj7 instead of D-7? In the same vein, what makes this rootless D-7 so special?
First thing which comes to my mind is that you get a (rootless) D-9 when you play FM7, so you add color to your chord (the 9th). And if you play the root in the melody, you don't "lose" a voice by doubling a note.
The second thing is that playing the rootless chords, especially when playing standards, change the feeling of the chord progression: playing FM7 / B-7♭5 / CM7 is a change from 'plain old' D-7 / G7 / CM7. It also gives you an opportunity to play with your left hand in a higher register and from time to time pop down the root in the lower register for an additional percussive effect.
Third thing is that you will get familiar with these voicings, which means that when you get to play with a bass, you can keep playing them without having to think about it and get the roots from the bass player.
In the end, it all boils down to a matter of style and how you wish to sound. My advice: try both on the same tune, and record yourself playing. Then listen back, and go for the one you prefer as your main style, keeping in mind that variety is the key. Being able to switch from rootless to non-rootless will bring more diversity to your playing which is always a good thing.
Depending on how much the audience is used to the type of music being played, it can "hear" the root although it's not there.
As if the player was saying "I don't play the root D there, but you know what I mean". The context helps pretending the root was there.
A bit like when someone is playing a very minimalistic version of a standard tune, if the audience knows the tune well enough it can hear chord changes that are not even played. Very few tones are needed to hear a blues progression, if you listen to a lot of blues.
There are a number of cases where it's possible to think of chords from different perspectives. One example would be the way that a C major 6th chord and an A minor 7th chord have the same notes in, but in different contexts it might make more sense to think of the chord as one or the other. Another example would be when modulating, say using a common chord - The I chord in G major is also the IV chord in D major, so you have to think of it both ways.
The idea of rootless chords, too, is a question of 'how you're looking at the notes you're playing' - rather than seeing the chord as just identified by the struck notes, you're seeing it as part of a different chord for some reason - either because someone else is playing the root, or because the root has been sounded in a previous phrase, or some other reason.
Alternatively you could see the rootless chord as a substitution for the normal, 'rooted' chord.
It's a matter of voicing: many chord changes happen by just changing few notes slightly. If there is a dedicated bass note keeping track of the respective root, it tends to jump around a lot, not maintaining a melodic line of its own. Omitting such a bass line makes the resulting changes more subtle and work on their own, like reciting a poem without making a marked pause after each line and stressing each stress, or showing a comedy without queued laughter or not explaining a joke.
Of course, there are differences in rootiness: the bass line for a Silcher chorale is a lot more bland than one for Bach: with Bach, you tend to have a reasonably solid melodic line for the bass without losing its fundamental function.
For Jazz, a walking bass line is also frequent and it just does not make sense then to try continuously providing an additional root for the right hand chord voicings.
