Regarding Starfleet captains staying on the ship or being on every away mission, I think one thing is being forgotten.
Archer, Kirk, and Pike make sense because of the era they took place... 22nd-23rd centuries. (Particularly Archer, as he was THE first Starfleet captain going out as far as he did.) It was a more frontier time... a space wild west. Keep in mind that TOS was pitched as a western in space, and with that comes certain expectations... like the lead (captain) being in the thick of the action all the time.
With the 24th century (Picard, Sisko, Janeway), space was not quite the wild west it used to be, so Starfleet regulations adjusted to that mentality. Which is why Picard rarely left the ship. Same with Sisko and Janeway. (Though they certainly left far more often than Picard, it was nowhere near as often as Kirk or Archer.)
Burnham seems to be the question here. By the time she became captain, they were already in the 32nd century. But she comes from an era where the captain would be leading away teams all the time. (We see Georgiou do it in the very first scene of DISCO.) She's following that example. But more importantly, the 32nd century is actually more of a space wild west frontier than the 23rd century... maybe even more than the 22nd. So while Starfleet may have adopted the 'captain stays on ship, XO goes on away teams' philosophy after Kirk's time, it's certainly possible Starfleet may have reverted back to a more relaxed view on captains always leaving the ship, especially since the times call for a more wild west mindset.