I honestly thought I couldn't possibly have anything to contribute to a thread 13-pages-long and counting, but having finished reading, I actually might, even if it's just recombining points already touched upon or implied by others.
The way I see it, but I also might have borrowed the distinction somewhere (Wikipedia?), there are two related but distinct meanings to the word "mode" in music:
-modes proper are scales that get to become foundations to a musical piece, i.e. the 7-note inventories from which pitches are extracted either in melodies or/and chords. A tonal piece is in major or minor mode; a piece based on a Dorian scale is in Dorian mode (its center of gravity, finalis or what have you has to be established by some means, like pedals or ostinatos, or employing functional harmony of the kind found in tonal music - I choose to call this latter approach para-tonal, but to others everything not revolving around the perfect cadence and a prevalence of progressive root movement is dubbed "modal" even if the mode in question is the Ionian. Fair enough I guess, I don't get to decide what's called what); even a scale played in isolation is in the mode that gets the name from that scale, because there is nothing else providing context;
-in the extended sense, "mode" is shorthand for "modal scale", which in turn means "Church modes other than the Ionian, possibly including the natural minor scale". Whenever you use a given modal scale in a context that is, itself, not based on said scale, you're playing a "mode" in this second sense.
Now: in a tonal and completely diatonic context, let's say major, why should you "think" Dorian when the ii comes up in the progression? The "mode" is just one face of the seven-faced die that is the intervallic system (two non-adjacent semitones and two groups, of two and three whole-steps separating them). If I just play said system, having learnt one face of it, namely that of the mode proper (I), and then favour chord tones placing them in strong beats and/or at the extreme points of stepwise movements, and avoiding the avoid-notes...I'm kosher. A possible added benefit of this approach is that I may be more fluid in my inter-chordal phrasing.
What happens if there's quick modulation, tonicisation, borrowed dominants or other chords, or whatever best describes specific non-diatonic phenomena? It depends: sometimes you want to think in terms of the new, temporary center, or else, again, rely on chord tones and alter the basic inventory, or rather the phrases you derive from it, inasmuch as chord tones do.
On the other hand, "thinking modally" has its own strong points.
-You can teach yourself the fingerboard more effectively, as said;
-it has undeniably useful expressive functions - modal scales have their own mood, so effecting a non-functional modification of the pattern over a given chord ("diatonic mode": by way of example, playing an augmented 4th on the I of a major piece and giving it a Lydian tinge) can spice it up, towards a specific, known, atmosphere. While knowing the effect of specific note alterations to the two main modes would also do, why not use already existing concepts? Nothing prevents you to, then, compare the "modes" and focus on where each "deviates" from the two, major and minor, used in tonal music as foundations and here assumed as "norm" - in fact, that's precisely what everyone does anyway;
-related to the latter scenario is the crucially important reason to learn and use modal scales: chord substitutions in jazz. (Disclaimer: I don't play jazz myself, so all or part of the following has a significant degree of likelihood of being inaccurate bovine excretion.) Given the possibility of replacing, beforehand as a composer or arranger or improvisationally as a musician with comping duties, diatonic chords with others of equal functional value, the melodic improviser has the choice of not following the substitution, carrying out a different one, or doing one when none is present in the chart, by outlining a chord with its tones or with its corresponding mode. Again, you can do without the latter if you have command of chord tones, but in this specific situation calling for a "vertical" approach, there are no negatives to resorting to "modes" as a mnemonic resource for improvisation.
Phew.
A note on the melodic minor. It is at its core a tonal concept: the seventh of the minor mode is/becomes a leading tone, and the 6th is likewise raised to provide a melodically easier approach to the raised 7th. I have no idea if Common Practice period composers thought in terms of "ascending melodic minor" or just "raise the seventh if the tonic follows, raise the sixth if a raised seventh follows". The thing is, the ascending melodic minor happens to be a different intervallic system as defined above: the two semitones are now separated by two groups of 1 and 4 whole steps. At this point, people decided to learn and use the other six possible scales derivable from this scheme: in short, do what was done centuries ago in musical cultures that decided to expand the expressive palette by incorporating scales different from the diatonic pattern, and use them as modes, in both senses of the word in our case. (Those cultures, by the way, probably precluded themselves from developing a functional harmony, for better or worse, precisely because they didn't stick, unlike we did, with a system pre-historically derived from the circle of fifths and thus allowing more consonant chords being created, first as a byproduct of melodic polyphony and then studied and systematised.)