Presumably you’d be sleeping in a tent, so the “sleeping outside in the rain for days” bit isn’t too bad.

How many days have you spent living in a tent in the rain? because personally I think it's kind of fun for one night, assuming it's not a complete downpour, and then the novelty wears off and everything is damp and muddy. And that's assuming you have a reasonably good tent that you've prepped for wet conditions, and you set it up in a spot with good drainage that hasn't become a puddle. Campfires are a lot less fun in the rain too.

Carrying everything is a pain though, particularly if it’s a cold season and you’d have to carry a bunch of furs to help maintain body heat while sleeping.

Yeah, cold weather gear, or if you can't rely on water availability and you have to carry it with you. Being able to keep yourself alive while not getting overburdened with gear is pretty difficult.

Only real way to do it in that case is to have an extra horse for cargo, which of course further complicates logistics

It really does. There was this excellent video that was shared on lemmy a few weeks ago:

What Medieval People Ate When Traveling

One of the things he covers (around 9 minutes) is the cost of keeping a horse while traveling (from historic records). Really great description of traveling considerations overall.