I don't know that there's a definite answer to this question, but I'll offer a few thoughts.
If we go back to the days of monophonic chant, as you said, Lydian was a scale that was like the modern major scale, with a flexible fourth degree (natural or sharp). Mixolydian was like the modern major scale, with a lowered seventh scale degree. (Note that Mixolydian chants rarely -- but still occasionally -- employed B♭, but when they did so, they could also be thought of as a "transposed Dorian." I mention this because there was really no strict delineation between "major modes" and "minor modes" in chant, as this division and grouping into two polarized major/minor categories didn't become clear until the early 1600s.)
To the extent that a "major mode" emerged in 15th and 16th century polyphony, it had the features you discuss. It typically had the major third above the final/tonic. It sometimes had a flexible fourth and/or seventh scale degree. The more frequent occurrence of E♭ in 16th century music allowed an F-final piece to perhaps dabble in the possibility of both ♯4 and ♭7 in the same piece, though that was rare. Sharps in general were rare and still regarded as musica ficta until later in the 16th century. Once they became more standard, and "chromatic" ideas reemerged with the new emphasis on rediscovered Greek theory, all sorts of accidentals started happening, leading to wide-ranging harmonies. (See, for example, the works of Gesualdo, or -- on a less extreme scale -- the Prophetiae Sybillarum of Lassus.)
But if we back up a few decades and look at the emerging sense of a cluster of "major" polyphonic modes in the 1500s, I'd say your description is pretty much on target. A lot of works will oscillate back and forth between a raised 7th degree as leading tone and a lowered one employed elsewhere (not at cadences), effectively the "Mixolydian" gesture. I don't think this gesture ever truly went away in popular music; it clearly survived in various religious and gospel styles and is present in modern popular music that employs ♭VII in various contexts. A lot of these bear striking similarities to many standard "major mode" works of the 16th and 17th centuries.
The ♯4, on the other hand, was transformed earlier into more of a leading tone to scale degree 5. It's more likely to occur in cadential gestures to that note in 16th-century polyphony, rather than as a melodic or coloristic alteration. That's not to say it doesn't happen, but my sense is that ♯4 gradually declined in favor of the natural 4 in the emerging "major mode," and the harmonies it supported gradually turned into modulations to the dominant. Of course, those eventually became the progenitors of the modulation to V that was standard in common-practice works.
There's part of me that would really like to think that ♭7 in Mixolydian was also the ancestor of modulations to IV. I think in some sense it is, but I think more explicitly it's the ancestor of the sort of "turn to IV" that frequently occurs near the close of a work in baroque and classical styles, where you'll start off a piece in I, go to V, then go back to I, and then the final phrase or two will have a ♭7 to hint at V/IV before the piece comes to rest. That sort of "plagal" turn toward IV has predecessors in final phrases dating back to the late renaissance, where perhaps (?) it was a remnant of the old Mixolydian inflections.
These last few paragraphs are kind of speculative, but if you take a hard look at Bach, one of the defining characteristics of "being in major" was generally a modulation to V. And while not as required, the little "turn to IV" near the end is quite common once you start looking for it.
To come to your articulation of this:
In terms of modern major keys, a raised fourth and lowered seventh are
used to tonicize the dominant and subdominant respectively. It's
noticeable those are the two variable degrees in Lydian and
Mixolydian. Can the two somehow combine to a major mode similar to how
minor mode has its own set of variable degrees? My first thought it
"no", because #^4 and b^7 change the tonic, whereas variable ^6 and ^7
in minor don't change the tonic. But that may be a hasty conclusion
from a largely modern, harmonic perspective.
I agree with your last sentence -- you're thinking about this a bit too much from a modern harmonic perspective. More specifically, I think you're approaching it too much from the lens of diatonicism, which was not the world that Bach or Handel lived in. They weren't taught theory with diatonic major scales as the basis of harmony for a year before someone allowed them to write a chromatic note (as many music undergrads are taught today). Accidentals like ♯4 and ♭7 were fundamentally a part of what "major mode" meant for them. Those were necessary and expected accidentals that were to be composed in at appropriate points in a piece, similar to how renaissance musicians understood that musica ficta was necessary to introduce a sharpened leading tone at a cadence.
So yes, I myself have frequently noted the fact that the ♯4 and ♭7 "modal inflections" seem to have survived as modulatory inflections in common practice music. There's more to the story, to be sure, but I don't think it's wrong to think of them as aspects of the "major mode" that were gradually transformed into harmonic/tonal/key elements.
I'm not sure if that answers your question. One last thing:
If the true difference in those modes lies way back in early Chant,
and the distinction truly was lost by the times of early polyphony,
that basic knowledge is helpful too.
It's really hard to talk about any "true difference" among modes, as modes are a moving target. Keep in mind that the medieval "church modes" existed in various forms for roughly a millennium before they completely gave way to the major/minor system. There was a lot of change and development over that time. (This is a common problem with thinking about the past, where the farther back in time, the slower we tend to think of change happening. But lots happened over the centuries, which was a much longer time for medieval and renaissance music than the interval between Bach and us.)
Thus, it's hard to speak of what was "truly" Mixolydian, as understanding of that mode in the 16th century was different from the first tetrardus modal concepts in the 9th century. If you really want to delve into this, I'd strongly recommend starting with the "Mode" article in the New Grove Dictionary, written by Harold Powers. It's quite long for an encyclopedia article, but it's about as concise an introduction to the manifold versions and concepts of "mode" that existed over time as you might be able to find.