Good points. I think playing in tune, like most basics, it’s good to start simply, like playing each note exactly the same way, as you suggest. It’s all about consistency and that can help, say, in keeping a constant embouchre even when playing a tricky part.
A counterpoint might be that our music imitates life, in that we speak in a constantly changing pitch, tone and speed; that’s what makes it interesting and holds the listeners attention. Balance, as always, is the “key”.

An update on another reason i found out recently why i’m still not playing in tune in songs even though i’ve now nearly been playing for four years.

currently working on some grade 6 material and came across 2 pages of exercises, which my teacher said a lot of students think these exercises are too easy, so they don’t bother to spend much time on them, hence one of the most common reasons why they play too sharp in the higher registers is because they haven’t spent enough time mastering these exercises.

The 1st exercise is just playing up the C major scale starting on Low C, but for every note you play in the C major scale, play it’s corresponding 5th note.
Ex if you play a Low C, then follow it by playing a perfect 5th which is Low G. Then a Low D followed by a Low A etc.. all the way up the C major scale.

But you must use a TUNER and make sure each note is pitch perfect followed by a pitch perfect 5th note. The hardest bit is playing the 5ths pitch perfectly! These exercises aren’t just about playing each note pitch perfect, its about being able to jump to a higher note (like a 5th) and playing that 5th note pitch perfect. If you can’t do these exercises pitch perfect, then you can’t blame your poor sight reading pitch on everything else like faulty reeds, mouthpiece,ligature, microphone, sax pads etc..

I’ve no problem getting individual notes played in tune, but i didn’t realise you also have to focus on getting pairs of notes in tune, as the second notes of pairs tends to be put out of pitch by the first note in the pair

I’ve spent 15 mins a day on this exercise for the last 2 weeks, and i’m still working on it. What i have noticed, if i just play the notes in the C major scale, they are now a lot better in tune with a tuner across the whole scale than they were a few weeks ago. What i have been told is if i work on the rest of these type of exercises, my tuning pitch would become good enough to play in ensembles. But what i think would happen, to go down that route one would have to warm up every day with tuning exercises ie resetting your in built tuner every day, to keep it in check.

one thing i have noticed, the more you work on a pitch, the hearing range of the flatness or sharpeness of a specific key does expand. whereas in the early stages you need a tuner to tell you if you are sharp or flat, later on, differences in pitch gradually start to be more noticeable.

the mind seems to naturally overestimate jumps in pitch, so i guess one has to reign it in and train it to imagine or expect a slightly flatter jump, before playing it.

The goal is to listen to your key pitch and recognise that sweet spot where the tuner needles says you are pitch perfect, while not looking at the tuner.

Here’s two interesting points David Liebman makes in his book ‘Developing a personal saxophone sound’.

Cork position

Overtone sound

It seems accurate tuning is controlled by the player and that the overtone produces the best tuned notes.

this is why one needs to make sure the Mouthpiece fits the cork, not any cork will do