Why are the Davison and McCoy era so polar opposite?

It's always seemed strange to me that the two eras, despite being under the same producer, are so drastically different in their tone and approach.

The Davison era was all round very staunchly serious, very apolitical, with the Doctor rendered a lot more fallible and prone to get it wrong or have to defer to greater authorities than usual.

The McCoy era is predominantly comical (or at least more willing to do frivolity), very political, with the Doctor now a formiddable master-planner, and there's a lot less of a concern of the show appearing 'posh' anymore, and certainly Ace seems to be an entirely different beast of companion to the Season 19 bunch.

Sure there might be some stories that cross-over between them. Enlightenment wouldn't look too out of place in Season 26. The Five Doctors and Remembrance of the Daleks seem like very close kin. But for the most part the eras seem to have nothing in common except ironically the guy in charge.

Yes a lot can happen in a decade, but quite frankly even the Pertwee and Williams era still seem to have more in common than not (much of the Key to Time season feels cut from the same cloth as Carnival of Monsters, Frontier in Space, Death to the Daleks, Planet of the Spiders), and bearing in mind the Williams era was made under the same kind of restrictions and heavy-handed BBC intervention against the show's previous violent direction.

It's strange that what seemed to be so staunchly defined as the direction for the show with Davison, was so thoroughly reversed three years later under McCoy. How indeed did we get so drastically from one to the other?

It's not exactly rocket science - one is Saward, one is Cartmel. Aside from suggesting when he wanted an old monster or guest character back, JNT pretty much left the story tone to his script editors.

Actually, I think that the drastic change in tone and style started in Trial of a Timelord. As you said, Seasons 18-22 were relatively serious, but even the pacing was still the same that had been built upon since the 70s. Starting from Trial, and especially in the McCoy era, the pacing became a lot faster, more stories had a balance between dark and gritty to the lighthearted and comedic vibes and even the cinematography felt completely different to how it was before, with Seasons 23-26 having more handheld shots, quicker cuts and even the cameras were more filming on film tape, which hadn't even been done, with the sole exception being Spearhead from Space. There's also more emphasis on music playing compared to past Seasons which were more sparse.
Also, there's far more explosive action and sparks everywhere compared to past Seasons. Take a shot of alcohol every time there's an explosion in Seasons 23-26. Actually, don't, I don't want to pay for your hospital bill.