Random extra one mile runs throughout the week, worth it or no?

You'll be fine. Go for it

I also think you'll be fine, but be careful about jumping on the treadmill with zero warmup. At least do some mobility first or do it right after you've been out for a walk or playing the yard or whatever.

If you are in need of some extra miles then it can help a small amount but I would note that I wouldn't consider it "quality training miles". Ultimately a 10 mile run is more than just 9 miles + 1 on the treadmill a few hours later, so as long as you are aware of that then go for it.

I would caution generally about this because the time you are running and before you are fully warmed up is one of the highest risk of injury periods (the other being when form breaks down and you keep pushing).

I think if you have 10-15 minutes around the day you will likely be better served stretching and doing mobility or foam rolling. It's a lot less sexy and doesn't show up on Strava, but it will likely help you out much more, especially if you are building up in miles.

There are a couple benefits, like increased calorie burn if you need to lose weight and overall higher mileage, but they won't be very quality miles, and my guess is they won't ultimately affect your marathon time by a meaningful amount. Your risk of injury will be slightly higher. Although injury risk for an easy mile on the treadmill is quite low, if you do it 15x/week all year, those tiny probabilities begin to add up.

You'd be much better off doing 3-4 extra 2-mile runs versus 10 1-milers. Or when you do find pockets of free time, hit the exercise bike, elliptical, or rower for 10-15mins instead - much lower chance of injury and gives you a break from running. Or do some strength work.

You aren't running extra miles, you are making up for deficient miles. One mile at a time is better than none miles at a time.