The post looks reasonably adequate as it was originally built even by modern standards. If it wasn't you'd have seen disaster long ago, when it first started to decay.
To maintain the style detail I'd rebuild to match, using pressure-treated lumber. By doing so you eliminate the need to cut that notch and you end up with a more robust post. A single post ...
There is probably a flush beam there that is installed up in the roof or floor system. You just hanger the joists into the flush beam instead of resting them on top of a dropped beam.
Example of a flush beam
Definitely add a header there if it's a bearing wall and you're removing studs.
Why is there no header for the full height doorways?
I can think of two possibilities:
The wall is structurally unsound due to the missing support and if so, you should see some signs of that. (i.e. sagging, drywall cracking, etc.)
The wall is designed such that the unsupported span load is carried by the support on either side of the doorway opening.