Attach rim joist to post supporting roof
It is mostly a style and preference choice.  The original half inch lumber post is probably not structural, though it could well have been a cheap replacement for one which was more substantial.  From what I can see of the eves, a 4x4 or 6x6 post is appropriate.  Some jurisdictions require heavier columns beyond a certain length, usually about 10 feet.  Raising a 6x6 post is not that hard with a helper or two, or straightforward solo job with block and tackle attached to the temporary support.

A properly designed and built deck should have no trouble supporting a post.  You could build the deck with an equivalent support which is not in the way:  slanted support, temporary horizontal truss, etc.

If you like the look of a stone or concrete base, it should not rest on the deck.  Build it into the ground (depth as required for freezing conditions in your climate) and finish the column on top of it.  Build the deck around the column&mdash;it does not need to be tied together and it is probably better there be a small gap (1 inch?) to allow for independent settling.

You could also support the roof with a 45&deg; to 60&deg; post fastened to the side of the house.