Let me just respond to each song example you’ve presented, and then I’ll summarize at the end.
Cliffs of Dover: The chorus seems clearly to me to be ii V I at the beginning. I highly doubt many if any theorists who specialize in rock music would analyze this chorus in Dorian. Sure, it’s possible, but I think that’s a mis-hearing (sorry). I’m not sure exactly what you mean by the “offbeat” focus, unless you just mean that tonic (G major) is not on the first bar of the four-bar phrase. If hypermetric downbeat-ed-ness were our standard, then pretty much every pre-chorus and bridge would be in a mode. That seems like an odd view, and goes against basic theoretical ideas about the integrety of keys and tonal unity.
Get Lucky: I probably wouldn’t analyze the song in F# minor, B Dorian, or B minor. I would choose A major and call it ii IV vi V. This example is kind of related to the “Cliffs of Dover” example, in that you seem to want to make the first chord of a progression tonic. But that’s certainly not a standard strategy. You may be concerned that “Get Lucky” never evinces the real tonic (A major), but that is a commonly-known issue in pop/rock analysis. (Mark Spicer gave a nice talk at the 2009 meeting in Montreal, which often gets cited.) It seems that your understanding of “tonic” is somewhat different than the commonly-held view of tonic in music theory. Tonic should be a stable sonority, with no palpable desire to move anywhere. It should sound like home. The first chord of that “Get Lucky” progression does not sound stable. It wants to move. That’s the great thing about that progression, in that it naturally keeps cycling back upon itself, in an infinite loop, because more stable chords (like F# minor) are in hypermetrically weak positions. I think F# minor as tonic is more viable than B minor, but I wouldn’t choose F# minor for the reasons that Ryan and I explored originally. In songs where it is ambiguous as to whether it is in the relative major or relative minor, it’s best to choose the relative major. That’s the way HookTheory was originally organized, and I think that is a good default. When I talk about charting songs in minor (the “minority” of cases, pun intended!), it is those songs or song parts for which the relative major has lost is perceptual pull (such as in Metallica songs).
Drive Hard: Here’s a song (at least what I can hear in this clip) where tonic is correctly identified. This song is in C minor, with the IV as mixture (let’s call this strong Dorian mixture, if you prefer). I have repaired one error you made in your transcription: that last chord is a G major chord, not another F major chord. So we have a major V chord at the end. Is it still Dorian? Mixture is just so prevalent in pop/rock, that using these modes seems somewhat trivial (or rather, happenstance or coincidental), assuming tonic is correctly identified. I find it frustrating to have to put V/i instead of just a regular V, since there is nothing unexpected about a major V chord if the tonic is minor. The fact that you have bVI but not bVII chords (instead, VII chords) due to the implied harmonies of the mode is very strange from a modern theory point of view.
Timeless: This is another example where what we’re really seeing is the great variety of mixture in a song, with the IV, bVI, and iv chords not conforming to any single mode.
The Simpson’s Theme: Here Lydian is manifest primarily (if not solely) in the melodic content but does not really affect harmonic structure in terms of chord progressions. (There really is no “chord progression” that I can see, at least within each section.) This is the same issue we saw in the Metallica example. As far as the modulations go, I don’t really have a problem with them. I do hear them as modulations instead of chord progressions.
To summarize, I think all of these examples of modal harmony can be better understood and charted through a more simple major and minor system that allows for mixture. We can understand the types of mixture that occurs through ideas borrowed from melodic modes. Indeed, I am fine with those types of discussions and have discussed ways to understand the mixture in songs through modal concepts above (e.g., the Mixolydian verse to “Hungry Like the Wolf”) and in my own academic papers. But, returning to my original point at the beginning of this long thread, using modes to determine the “diatonicness” of certain harmonies seems misguided and has been abandoned by the music theory community. You cite Nicole Biamonte’s 2012 talk on modal mixture in rock and metal in your post above. This paper only further proves my point. Nicole sees the Doors’ “Break on Through” as evincing Aelioan qualities, but she labels the progression not as i-VII but rather as i-bVII, i.e., with Roman numerals referenced to the major scale. Similarly, her Phrygian example of the verse to Iron Maiden’s “Remember Tomorrow” is labelled i-bII-i-bVII, again using the Roman numerals calculated from the parallel major. It is worth pointing out that with this example, bVII does not strictly belong to the Phrygian mode, since the third of the chord (a regular scale degree 2) is not within the Phrygian collection. So even in the handful of very short examples she provides (and this is not the only example of hers that shows this problem), the modes themselves are poor ways of accounting for the harmonic events.
In summary, theorists still use modes to refer to harmony, but we do so primarily as a convenient way to describe the common types of mixture found in minor and major keys. Dorian is a major IV chord in minor; Phrygian means using a bII chord in minor, Lydian means a major II chord in a major key; Mixolydian means a bVII chord in major; and Aeolian means the natural minor collection, involving in particular bVI and bVII. But analyzing entire swaths of songs “in” a single mode and then having departures from that mode being modal mixture within a mode is just making things unnecessarily complicated. If I write the progression i-bII-i, I have no problem with someone saying that it is a Phrygian progression, but I am not going to analyze it as “in” Phrygrian and then adjust all of the rest of my Roman numerals based on the song being “in” this mode.