If you A/B an Ehrlund EAP and a high quality microphone, you will hear the difference. One will sound like a pickup and the other will sound like the bass without pickup. My experiences with Ehrlund and Schertler (the pickups that attach to the body of the bass with putty) is that you really have to experiment to find the "sweet spot" where the sound is clear and not too complex with overtones. I'm not saying pickups are bad, and I've had success mixing a tiny bit of pickup sound into a mostly-mic sound on the bass. I've also made nice recordings with "only pickup." There's not one fits-all solution for every situation, but after years of recording and trying to get a bass sound that works, this is what I've come up with
We've used countless setups over the years. In recent times since we're making more videos for YT and Facebook, the recording team and the video team are trying to get the best recorded sound as well as a good-looking video. Then came Corona, and we had to play with huge distances and spread-out stage setups. Sometimes it's a compromise as to what we can all hear live or through monitors, and what the video crew likes to see.
You're right—the Schoeps Mk41 is a supercardioid. That means it picks up in an extremely narrow pattern in the front of the mic, but there is still a pattern that picks up in the back end of the microphone—which is why the mic shield works to keep those sounds out.
I use the amp in rehearsal and live situations since some players in the band do not use headphones. They want to hear the bass in the room. It's a catch 22 because the recording technicians are trying to isolate the bass, but the horn players want to hear and feel live bass. I use the amp mainly for the other musicians to get a live blend and to be able to hear the bass even if they are not using headphones. I use headphones mainly to hear the piano, which is also isolated quite a bit. You don't hear the bass amp through the bass mic because I always set up slightly to the side of the amp. It's not pointed into the bass mic (and not into the back of the bass, which is also problematic).
Hans has a quite normal drum setup, also with the cymbal height ... similar to Mel Lewis, Peter Erskine or Jeff Hamilton and many great big band drummers. He plays very dynamically, so the loud parts are loud and the parts with woodwinds, or solo bass, etc can be heard without artificial support. We've had lots of great drummers as guests with the band and they generally do not "bash" ... it's a fallacy to think that just because it's a big band, the drummer has to play loud. The band—including the drummer—should play dynamically.
This arrangement was written by John Clayton, who knows how to write a bass feature without the band getting in the way. As you point out, there are moments of very loud, ff playing (the intro where the bass is playing arco, then the shout chorus at about 3:54 where all of the horns are loud).
Sorry ... I don't have the isolated tracks! Thanks for checking it out in such detail though @Wasnex
Thought I posted this yesterday

facepalm
I want to clarify one of my points. IMHO, the Ehrlund EAP sounds closer to a mic than other pickups I have experience with. I agree it's not 100%, but it's pretty darn good on some basses...not so good on others.
You are correct, you have to find a sweet spot on the bass for the EAP. The person who bought the EAP spent a couple of hours with a stethoscope and assistant.
Something I did not mention is bleed from other instruments happens with a pickup as well, but the ratio between the bass sound and the bleed is usually significantly higher than with a mic.
A mic on an upright in a live band situation often does not produce a nice authentic upright sound. The main problem is the upright bass is not very loud relative to the other instruments in a big band. The bass playing solo through the mic may sound fabulous, but when the band kicks in, the bleed from other instruments is often louder than the bass, and those instruments may not sound very good in the mic signal. IMHO, at some point, the bleed from other instruments through the mic tends to become more of a problem than the tone degradation caused by the pickup.
Another problem is the upright bass generates different frequencies and impulse response envelopes from different positions on the instrument. So you generally do not get a perfectly authentic sound by focusing the mic's pickup pattern too closely on one spot. This relates to using a very directional mic almost touching the body of the instrument.
In our case the bleed from the cymbals, in particular, made the signal unusable (IMHO), but I was not the person who made the call, and we used the mic for an extended time.
We used rather tall plexi drum shields. I don't remember if the particular drummer had high cymbals or low. Even with the level of bleed from the cymbals, the signal may have worked fine. But the cymbals were very harsh and phasey sounding. I believe this is because of the mic technique we used on the bass.
We used an AMT S25B mic mounted on the bass with some sort of clamp. The mic was placed very close to the face of the instrument. I believe the offending cymbal sounds were coming in the front of the mic after being reflected back off the curved face of the bass. Because the cymbals were too loud and distorted in the mic signal, I felt it was a lost cause.
The other problem with our attempt is the bass player tried to blend the mic with a pickup and send the signal to both his stage amp and FOH. IHMO if you going to use a mic and an amp in a relatively loud environment, I strongly suggest keeping the mic out of the amp, and also any nearby monitor wedges. The mic will tend to feedback through an amp at a very low volume in my experience. You might fill out the lows a bit, but if you need the amp to actually increase your stage volume, it's probably not going to work very well.
I don't want to come across as being dogmatic. IMHO there should be sort of a balancing test between the advantages and disadvantage of using a mic VS the advantages and disadvantages of using a pickup. How these factors are weighed should vary somewhat by situation. For example if the priority is live sound and the stage volume is high, at some point the mic will become unusable, or least it's not of any real benefit do to sonic degradation. If on the other hand, the priority is recording and the stage volume can be managed, then the mic can often provide better results. You are obviously getting good results, and apparently using every trick I can think of, if not more.
I would caution people to not generalize too much from your success. Yes a mic can be used successfully, but it's not a matter of simply putting up a mic and everything is perfect. Getting good results requires certain knowledge, skill, technique, resources, and compromises. It also requires experimentation to figure out what works, and continuity of knowledge once the necessary trick are learned. So you may get consistently good results with one set of musicians and sound crew, and bad results if either the musicians or sound crew are changed.
When it comes to dealing with "A listers", the correct answer is often "Yes Sir"...and you accept/manage whatever compromises occur as a result. You use every skill, tool, and trick to get the best possible results. If you running sound for an "A lister" hopefully you have everything you need to get a good sound.
For the record, I have never used one of the wrap-around mic shields. But I have frequently suggested that it
may be a helpful tool

.