First you need to check your local building codes. Any alterations will have to meet the existing codes; what you have is grandfathered in UNLESS you change. It is some pretty complex geometry.

https://buildingcodetrainer.com/residential-stair-code/

If you want more depth to your treads, you need more run. It appears you have about 18" more run before you'd have to move your doorway. You'll want an experienced stair builder. I doubt he can rebuild these stairs cost effectively.

agree. sometimes short stairs are used deliberately to fit a staircase into a shorter distance, and you might end up crowding the door at the bottom of the stairs. you will also probably be shocked to find out what staircase surgery costs. but have somebody look at it for you!

you need a professional stair guy. (not a gc) there are very strict codes for 'rise/run' of the actual treads and very little room for errors.

I learned this the hard way 10 years ago. call in a pro and have it done correctly and to code. If not, it could come back to bite you in the rear when it's time to sell.

The solution to a longer tread run will be to change the landing to a winder step. basically cutting that step in half diagonally. That additional riser will make all the difference. Yes, it will require a good carpenter and the stairs will need to be completely rebuilt. 1 day demo, 1 - 2 days frame, 4-5 days for finish. Plus plaster and paint. Depending on where u are in the country and finishes this is a $4k-$12k+ project, but would also give you an opportunity to upgrade that old style railing.