I appreciate the time and effort you put into this question -- and it took some time to get through it, and I'm not crazy about doing that at the expense of trying to answer more and more questions and get to a lot of different topics. Allow me a chance to offer an answer, but I'll start with a refutation.
Those are not the two philosophies in hitting.
Batting average has long been minimized as a telling stat, regardless of what teams might say publicly or how they might reward a batting champ each year based on batting average. OBP has been for several decades now the more valued number, so has SLG, and in the end it's the fusion of the two, OPS, that has guided so many teams when designing and ranking and building their offense over the past two decades.
It's basically this: A measure of who makes fewer outs, and what noise they make when they don't make an out.
That's the philosophy of offense that is spelled out by most teams this way: They want to find hitters who hit the ball hard and hit it hard often. Those two things in concert lead to what they call "damage" and the goal is to have a greater OPS.
There are analytics that measure the value of getting your star players more at-bats with runners in scoring position. That is both the OBP and OPS of the players ahead of them as well the lineup dynamics that teams talk about and many people study. Shildt calls this a "holistic" approach to a lineup, and he's touching on what you mean by a blend of power and average. But it's not that. It's a priority of a) not making outs as a hitter (OBP and average) and b) doing damage with hard contact (SLG and OPS). (Aside: Launch angle is PART of the equation; we over simplify the discussion when we fixate on that. Yes, more damage is done in the air, so that's why it's sought.)
Talked with Jeff Albert recently and he hit on something that you're getting at, too, I think. He made the point that a single counts twice in OPS. A double counts for both a hit toward the average, a time on base toward the OBP, and both two bases and a time on base for OPS. So, naturally doubles and singles and average are valuable. Average is still a representation of a good hitter -- look at the top average hitters in the majors -- but that's because it's a tagalong stat. Show me a great OPS and you'll see a great hitter, no matter the average.
There are currently 17 players with an OPS of 900 or greater. Of that group, six have averages greater than .300.
Fifteen have a batting average greater than .270 and that includes Ohtani.
The two that don't are Joey Gallo, batting .235, and Kyle Schwarber, hitting .253. Gallo has a .394 on-base percentage. He does not get out nearly 40 percent of the time -- or two times out of every five times he goes to the plate he does not get out.
That is 50 points higher than the highest OBP in the Cardinals' lineup.
Wouldn't you welcome Gallo in the lineup because he's getting on base and not making outs at a clip higher than any Cardinal hitter, and when he does hit he does damage, so that .235 average is misleading. I hope that this helps, and that you consider that hitting philosophies are not as divided as you think. They are mostly pointing in the same direction, and average can tag along, but it's not the north star, to borrow a phrase from the Giants' hitting coach.