Once thing I notice in watching videos on youtube of small jazz combos performing for TV, was how physically distant the performers are from each other, like many meters apart. It makes using microphones a lot easier.
It also helps if the drummer has the aptitude and attitude necessary to play in a jazz combo. In a lot of the videos I see of jazz trios, the drummer is playing with what many would consider a very, very light touch.
I was looking at some of the KNKX public radio videos with the Christian McBride trio. It looks like they are consistently using an EV RE 20 on Christian's bass. The mic is actually not really that close to the bass, and the RE20 has a really loose polar patter, meaning it picks up a lot of bleed. On the plus side, I think the off-axis response of the RE20 is relatively uncolored, so bleed from the drums and cymbals would not be to problematic as long as the sound from the bass dominates. Note: some of the videos have a SDC taped to the RE20. I believe the way this was done would tend to make the RE20's pattern more omni, because the tape is blocking many of the mic's ports.
I like Christian's tone in these videos quite a bit. Although he makes it look easy, the amount of string excursion produced when he plucks suggests he is using a fair amount of strength.
Compare to this video.
I believe Christian is running an AMT SB25B on the F hole for his stage rig and the FOH has an AKG C414 off the bridge.
I would say the sound of the bass is pretty good considering Christian appears to be running a mic into an 810, and FOH is using a second mic. Also the drummer appears to be hitting a lot harder in this video, and it is a huge acoustic space complicated by the needs of live sound reinforcement.
I think the results would have been better with a pickup run to a smaller amp, and FOH taking a feed from the AMT, but that's just an educated guess.

! Subjective assessments are funny like that