"In this philosophy, the mode was defined by the MIDDLE note of the tune, not the endpoints of the scale. Melodies were usually constructed with this as a "home". Melodic phrases generally started on this note and either resolved to 1/5 higher (or 1/4 lower) - an "open" phrase, or resolved to the home note - a "closed" phrase."

"MIDDLE note of the tune"

Here I ASS-U-ME that you REALLY mean 'MIDDLE note of the pitch RANGE of the tune'

Hmm, I'm no longer a teenager, so I'm no longer an expert, but what I have learned was that it was the 'ending note' that defined the Mode. Some tunes did start there too.

"the mode was defined by the MIDDLE note of the tune, not the endpoints of the scale."

This sounds more like the 'Plagal Mode', not the 'Authentic Mode'... the latter from which the 'Major & Minor scales' evolved.

Part of the 'Musica Ficta' game involved knowing just when you flattened a (usually seventh) pitch, and being not confused by mixing 'Authentic & Plagal' Modes.

Earlier real 'Modal Music' was strangled and mangled to fit into the new evolving 'music' based on the 'King of Instruments' - the keyboard, as has been mentioned above.

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"Dorian mode is recognisably itself whether you render it in equal temperament, Pythagorean, the classical Turkish system, third-comma mean-tone or whatever. "

I'm sorry, but all I can say is 'Humpty dumpty strikes again!' - I'd really like to see and hear a demonstration of this thesis - and it can't be in midi - my (normal PC) midi player can only play in Even Temperament! The basic 'Tone - Semitone' spacing arrangement might be DEFINED to be 'itself', I suppose, once one has got one's ear 'calibrated' for each of the differing microtonaly tuned systems... but of course you have compared apples with oranges, because the technical BASIS of the definitions of 'Tone & Semitone' is redefined (look at the oscilloscope and measure the frequencies!) in each system you mention...
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"many instruments, such as violins and trombones, are not tempered"

Sigh! If only THAT were true - you can play in any temper you want on those, you just have to blow, or finger the 'naturally occurring Just' notes to the even temper... See "Even Tempered Trombones", is an OLD Brass band Players JOKE! above.. you can even play in a bad temper if you want... sorry.. :-)
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If you are playing with modern 'Even (1/12) Temper' instruments you have to discard all thoughts of 'Ancient Modes', and start all over again, generating 'New Modes' which are really just using 'old words' for 'new concepts' that are similar and perhaps related, but NOT exact copies of the 'original concepts'. Humanity does this confusing bungle all the time, and Politics is Absolutely Trumps at that...
:-) (Un-American Activities, Patriot, etc, indeed! Hahaha!)