Can one use rootless chords along with rooted chords when constructing common chord progressions.
tldr; yes, this is okay both in group settings and in solo jazz piano settings.
Playing with a group
New jazz piano students may be told to use rootless chords when playing in group settings to avoid clashing with the bassist. This is good advice in general (and if you play a lot of bass notes, the bassist may give you a look). But you're not prohibited from ever including root notes in your voicings--even in the lower register of the piano. Bud Powell used a 1-3 left-hand voicing and a 1-7 left-hand voicing. Thelonious Monk frequently used a 1-5 left-hand voicing in the lower register while playing with a bassist. McCoy Tyner did this too, and it's a good exercise to listen to what the bassist is doing during those moments. All of these techniques have remained pretty common among jazz pianists. (Oscar Peterson also had a tendency to play a lot of roots in the lower register in his trio recordings.)
Additionally, when the one-handed Type A / Type B "rootless" voicings get extended to two hands, it's common to hear roots sneak into the upper register. This is also true of upper structure triads, some of which contain the root (e.g., when alternating between the bVI triad and the bV triad in the right hand and playing the 3 & 7 in the left hand over a dom7alt chord).
In solo jazz piano, it's also okay to play a mix of rootless voicings and voicings with a root. It's true that, in this context, the pianist must fill the role of the bass player. But the pianist (a) doesn't need to play the root in every voicing, (b) doesn't even need to play it at the start of every measure / on beat 1, and (c) can not play the root altogether in some measures. In general, Kenny Barron and Bill Evans provide good examples of how to strike this balance. Bill was especially skilled at delaying the root until after beat 1. For a more extreme example, you can also check Chick Corea's intro to Now He Beats the Drum/Now He Stops, which features a lot of quartal voicings (based on 4ths) and omits the root more often than includes the root (check out 3:18-3:52).
For a start, define 'rootless chords'. Is viidim a rootless dom7 or a chord in its own right? How about iii? A rootless Imaj7 or...?
And what's a 'common chord progression'? I guess it's more usual to use rooted chords.
Of course, jazz pianists often play rootless chord shapes in the expectation that a bass player will provide the root. They aren't rootless overall.