OK, so basically there is no magic bullet that I'm just unaware of. So fabrication: The piece that abutts the tile should be level with the tile and the thickness of the tile and thinset. The part that will cover the wood floor will receive the rabbett. If the tile and thinset is roughly 1/2" and the wood is slightly lower (maybe an 1/8th), about how overall thick should I make the piece? 3/4"? Also, would you fasten it with a bead of adhesive then nail and putty? Thanks

I'm the OP. Those marble strips are nice but they only come in white. Since our floor is dark they wouldn't work for us. We ended up going with wood. There was no molding of the right shape or height, especially since we were trying to bridge a height transition: the tile being higher. We bought ranch molding that was designed for window and door frames. If you are not familiar with ranch molding, in profile it has a slope: one side is high and the other side is lower. The installers has to shave some wood off the back to make it fit (to reduce the height), but that was their problem since they had promised they would come up with something nice for us and had not. We had them bring the wood right before the weekend and then we stained and varnished it to match our other moldings. On Monday they came back and installed it. The carpeting is a berber, and the edge is tucked lightly under the lowest edge. The high side butts against the tile/durock and a small caulk line is in between.

I see two options. The first is to have your tile person straighten out the edge of the tile. Fabricate a reducer that's 1/4" thick i.e. the difference between your wood and tile), finish to match the floor and cut it to length. Undercut the door stops to the height of the tile so the reducer will slide under. Fill the gap between the wood and tile with something that's the thickness of the wood. Apply a little construction adhesive to the bottom of the reducer and install it, weighting it down until the adhesive dries. This can be done by closing the door and sliding wedges under it. The second option would be to use a matching baby threshold that would lip over the tile and hide the uneven edge. Fit it, fill the void, undercut stops and secure with adhesive. Trim the door bottom if needed. The bottom of the baby threshold might have to be shaved off so it sits level. If it's made for a 1/2" floor and your difference is only 1/4" you'd have to shave off 1/4". The first solution yields the lowest stepover height but is more work. The second is easy but has a bigger hump at the doorway.

Not sure I'm following with the "age of the hardwoods" issue, but I've never seen the traditional wood tile look anything remotely close to actual hardwood (grout lines, repeating patterns) - but one alternative: they make some interesting parquet type wood tiles. https://www.tilebar.com/barberry-decor-nocciola-24x24-matte-porcelain-tile.html?gclid=Cj0KCQjw1PSDBhDbARIsAPeTqrfwSflz_oEHC6XQTPXeJvsJm_UWSWm355e1emzHFzt5DxvLVZ6ZJNsaAumiEALw_wcB&gclsrc=aw.ds Depending on the style of your home, if you found something like this in a similar stain/color, it could work.