Solid Hardwood, Engineered and Laminate Flooring - hardwood floors at entrance of condo unit

I just moved into a condo where they had hardwood floors installed all the way to entrance of the condo unit, right up into the hallway. Prior to that, it had some ceramic tile at the entrance, aluminum weatherstrip right under the door, and then carpet inside. The hardwood runs under the main entrance door and extends out another 4 inches or so, maybe an inch beyond the door frame.

This area gets really dirty. The floors are probably 10 months old now, but around the door, you can see several dirt stains that do not come off with traditional cleaning and more dents and bruises due to pebbles and stones. I wanted to put some type of doormat directly inside the door, but there is not enough clearance for the door to swing over the mat, so the mat needs to be placed 2.5' further inside. This still leaves a good amount of flooring exposed, which still gets dirt, sand, and water from incoming shoes.

I would rather not rip out the floors here and have tile installed, although that would probably be the best long term solution.

The only thing I can think of is to use vinyl floor to cover the area directly behind the door. It would be thin enough so the door could open over it. But I don't think it's going to look good, with wood outside the door, then 3 feet of vinyl, then wood again, and I will probalby place a runner rug inside also.

Anyone else have this type of setup, and what do you do, or anyone have suggestions?

Also, since the weatherstripping under the door is gone, I'd like to find a solution for that as well.

TIA