Pianotech

I would not call that a fixed pitch instrument. I asked my son how often he tuned his trumpet, after he asked me how often a piano needed tuning. He said every time he played. Came back in the room 15 minutes later and said he tuned every note he played.

Clearly, that trumpet needs 'fixing'.  

As for the sax player's comment: "...that the instrument was much more in tune with him than a normally tuned instrument", I fear that has to be taken somewhat in context  of the first few bars of the video, as well as a real-world understanding of whose role it is to make such adjustments.  Listening to the pre-performance tuning (piano and sax) would have been illuminating.

David P. You're getting off on the wrong foot here. Most percussion instruments are fixed tuned like the piano. Nearly all other instruments, wind, strings, voice, etc. are not fixed tuned. The musician is responsible for pitch and tone from the beginning through to the end of each individual note. Furthermore, none of these instruments are built to play in a tempered scale, all instruments are made to be internally in tune with themselves, they have to be for obvious reasons. As these instrument's overtones are fairly close to 'pure' they are tuned accordingly when they are built. However, non fixed tuned instruments have a broader range of variation whereby the musician can exercise control over both the fundamentals and partials. While we piano tuners concern ourselves with increments of a few cents or even fractions of a cent. The other instrument families have an almost infinite range of variation. In your first video you can hear the saxophone player quickly make adjustments to match the intonation of the piano in the first few bars. If one listens to to Grieg's piano concerto, in most performances or recordings, after the opening piano solo, one generally hears a big crash within the opening chords as the orchestra searches and finds a way to accommodate the tuning of the piano. While a UT will have some pure intervals that will match a typical orchestral instrument, it will also have some distinctly non pure intervals that will not be a natural fit with an orchestral instrument, so it's just as problematic as ET in that regard. 

The musical idiom also plays a role as different musical traditions utilize different values regarding pitch and tone. Expert jazz saxophonists for example are able to bend each note of the scale a full half step below and above the centered pitch. Instrumentalists are able to vary the coloration of the partials for further expression as well. 

There's no need to shoehorn your ideology into this inner world of orchestral instrument intonation. Orchestral instrumentalists have the ability to adjust their intonation on the fly to meet whatever contingency is necessary on a note by note basis. For orchestral instruments on a practical level, the idea that they are constrained to only 12 pitches while playing Western music is just a short hand, as I say, the actual intonation varies according the contingencies of the moment. The mindset that a musical scale is limited to 5, 8, or 12 discreet pitches is one that pertains to the world of fixed pitch instruments but not the rest of the acoustic musical world. 

If you want to learn more about the methodologies behind intonation and technique for non fixed tuned instruments, I highly recommend FINDINGS: MY EXPERIENCE WITH THE SOPRANO SAXOPHONE by the jazz virtuoso, Steve Lacy if you can get your hands on a copy, it's out of print but is considered a masterwork detailing practices for playing the sax with much time devoted to intonation. 

Back in the days of old, when I was an analogue recording engineer, I learned from studio musicians, notably string and horn players, that playing concerts, or just doing studio session gigs, that featured a piano were not their favorite gig. All non piano musicians, regardless of the instrument they play, learn how to bend notes to stay in tune with the rest of the group, but having to bend notes to stay in tune with the piano is a lot harder. Since the piano is really the only instrument in the group that you can't really touch up the tuning, or bend notes during the performance, all the rest of the musicians in the group have to do the compensation for whatever temperament the piano was tuned to. Unless a performance group is all one kind of instrument, there's really no way around this need to compensate, or bend notes, in order to play with other kinds of instruments. It's just part of being a musician in a mixed instrument performance and not any instruments fault. 

One of my influential memories was in my senior year of high school.  I am from a family of piano tuners and musicians and my high school senior year was spent at a magnet school where I was majoring in music.  In a music lab I heard a recorded performance of a university saxophone choir that played in perfect tune.  I had never heard perfect tuning before and the sound of 7 musicians playing without equal temperament was life changing.  I love to hear chamber music and a cappella choirs and often recommend that customers try listening to well trained choirs and chamber music on YouTube.  Our finest tunings are actually poor in comparison.

Sax is pretty far from being "fixed pitch." Any sax player can and will bend pitch within a much wider range than most other wind instruments.

That said, "melody instruments" (instruments that play one note) are constantly bending pitch to meet various aims.

For example, they might bend a note sharp as a leading tone, or they might bend a note flat as a "blue note" to meet expressive melodic goals. 

In ensemble, they will bend pitch to blend harmonically with the other instruments. 

Only fixed pitch instruments play in fixed temperaments. Fixed temperaments are all compromises, trying to meet the needs of the music is a "least bad" way. 

For the piano in particular, an equal temperament that is based on maximizing the alignment of partials throughout the range will lead to a coherence of sound, reinforcement of every note when the dampers are raised. This is what pure twelfth or pure 19th tuning is aiming at, and is what PianoMeter and PianoScope do - and I guess that OnlyPure is aiming at that, but its methodology is too obscure to make any real assessment. How a piano is tuned is important for its voice as an instrument.

For non-fixed pitch instruments, it can be a problem to align with the piano or any fixed pitch instrument, but that is the nature of the problem of tuning. The best tuning for an ensemble is one that is created on the fly, from note to note, from chord to chord: the pitches aren't fixed, they adapt.