From the discussion on the video link:

"There is a misconception that Bach supported equal temperament because of his famous 1722 collection, Das Wohltemperierte Klavier (The Well Tempered Clavier). Bach's clavichord was not equally tempered, but well tempered, like the title of the work says. The 48 pieces, two in each major and minor key, were written to show the character of each key in this temperament, the effect being completely lost in equal temperament. One can experience this by tuning a keyboard to well temperament, and then transposing Bach's Prelude and Fugue in C-major to C#-major, and vice-versa. The results will clearly show what the master was up to, and that the pieces were written for the nuances of each key with its particular coloring. Bach's theoretical framework was for pure intonation, and his music was written to be played in pure intonation, either by altering the tuning of the keyboard for each piece, or by using a flexible temperament that allowed pure tones in most popular keys."

"Equal temperament, a tuning which disfigures the natural intervals of the harmonic series, was invented as a necessity to the mechanical and engineering limitations of Middle Age instrument builders. But now we are in the 22nd Century and we can accurately measure the 1000th part of a millimeter. So why don't we use this technology to create in-tune musical instruments?"

I strongly recommend to read that discussion, it has both pros and cons of equal temperament and people are discussing the issue from both sides.

That's incorrect.

Equal temperament is very recent. Medieval through Baroque used a system of temperaments called meantone that began with C major being in tune, and then distorted tones slightly in order to make the proximal keys usable. Choosing a key in meantone implies a real difference in what the relative distance among notes was. When you go out beyond three sharps, it becomes very distorted. Five sharps is essentially unusable in meantone systems.

Bach's well tempering was a replacement for meantone where all the keys were usable. We don't know exactly what it was, though there have been some interesting attempts at reconstructing it. I remember being in the orchestra for the Bach four harpsichord concerto as a demonstration of one such attempt, which was wild.

As for using technology to create in-tune musical instruments, the original one, the voice, already has that. For keyboard instruments, there have been all kinds of crazy ones built over the centuries that have many more notes to an octave to allow much more precise intonation. But that's not the core restriction. I mentioned elsewhere in this thread that the pitch a given note maps to changes during a piece of music if you don't have tempered instruments involved, and those changes are always compromises.

So I wouldn't suggest reading that discussion, as it's apparently full of inaccurate material.

Bach wrote for meantone and later well tempered. Mozart was working with further variations on meantone, as was Chopin. Shostakovitch worked in equal temperament.

Shostakovich is a 20th Century composer, died in the 1970s, so at least as far as he is concerned, he wrote music with today's tuning systems.