You should hit the root at the first beat. That's what the bass is for. Then the solo instruments can hit the chord notes. If you skip the root on beat one, do it on beat two.
Uh, no. A good jazz keys or guitar player will be throwing chord substitutions / inversions left and right. If you are making the rookie mistake of sticking to the Real Book 1 chords, a more experienced player will tell you which chords are wrong, but you will see that for the most part both chord versions would work. For instance, you may come across a iii / vi / 11 /V7 progression. First time the keys guy will play it, at least behind the head, but when it comes solo time, you can bet he will throw in a chromatic descending progression from the iii. Or change his chord inversions so that, (using the key of Bb) all of those chords have Bb as the low note. That is iii = D minor #5, vi = G minor 7 1st inversion (3rd on the bottom), ii= C minor 7 with the 7th on the bottom, and V7 sus with the Bb on the bottom. Why? to get a pedal tone effect. If you insist on walking all 4 chords, you will mess up the pedal tone, so you should have your ears open and pedal the Bb with him. Plus there are plenty of jazz chords where the root of the chord is not what the bass should play- Bb sus with Eb in the bass, C maj7 with D in the bass
Another example - Cloud Burst. Fast = 318 BPM. An ascending / descending chord progression from Bb to Eb and down again. Then a ii / V7 / 1 in Eb, then the same, in F. So what is that, actually? It's noodling around on a Bb, then play the IV chord, the the V7 chord. Well, a simplified explanation. If the keys guy plays root position chords the tune sound clunky. But if he plays a series of inversions ( root, 1st inversion, 2nd inversion, etc) for the verse then you get what the song is about.
At one point, I was reflexively playing a ii / half dim ii / 17, every time I saw a ii / V7 / 1 (this is based upon using tri tone substitutions to create chromatic chord progressions. And it works in reverse). You see a 1 for 4 measures, you can play a iii / vi / 11 / V. Just make sure you hit the 1 before you go to the next chord. That's where the fun is - your lines have to make sense - they have to have a direction and flow, so you have to figure out what chords can be substituted and create on the fly a smooth walking bass line, all the while cueing off the other players. You can't abdicate your role as an anchor, but OTOH you can't be the guy that just walks through the stock chords all the way through the song. Always, before you play a tune, analyze it for substitutions and inversions.
To be contrary a bass player has to know theory and chord substitutions. You do not have to hit the root all the time, but you must be aware of points of resolution places where you anchor the song with a tonic or an obvious chord progression like a 11 / V7/ 1. I, mean, this is jazz and jazz is an exploration in real time of harmonies. My approach is to keep it to the agreed upon chords for the head and tail, but throw the substitutions when cued by the soloists. Keep your ears open, and if the ensemble starts to feel shaky, that's when you go back to basics.