One of your concerns is (let me quote from your question)

Often I read that there is the possibility of having a family X1,…,Xn of random variables on the same space. I know no example—and would be happy to discover—of a problem truly modelled by this, whereas in most examples that I read there is either a single random variable

Here is what I do on the fist day of my pobability class.

The statistical experiment I describe is: Go to the road outside the college building and consider the first car that goes left to right after your arrival. As we do not know/cannot predict which car in the city might be there it is a statistical experiment. The sample space is the set of all cars in your city (or in your country).

Questions:

1. How many people are in that car?

2. What is the amount of petrol in the fuel tank at that time?

3. How many kilometers the car has travelled that day before you noticed?

4. What is the wavelength of the color of the car? (admittedly artificial)

All these are random variables on the same sample space.

Answer to question 1 might be useful to a person who sells eatables on the roadside? (more passengers means more business)

Answer to question 2 might help decide if it would be profitable to open a petrol-selling shop here.

I ask students to come up with examples of such statistical experiments instead of coin-tossing throwing dice.

I got this from a bright student:

Go to the library. Observe the first book that is borrowed by a user that day. Sample space is all books of the library.

Random variables are: Number of pages of that book, Price of that book, How many times it has been borrowed earlier.

One of your concerns is (let me quote from your question)

Often I read that there is the possibility of having a family X1,…,Xn of random variables on the same space. I know no example—and would be happy to discover—of a problem truly modelled by this, whereas in most examples that I read there is either a single random variable

Here is what I do on the first day of my probability class.

The statistical experiment I describe is: Go to the road outside the college building and consider the first car that goes left to right after your arrival. As we do not know/cannot predict which car in the city might be there it is a statistical experiment. The sample space is the set of all cars in your city (or in your country).

Questions:

1. How many people are in that car?

2. What is the amount of petrol in the fuel tank at that time?

3. How many kilometers the car has travelled that day before you noticed?

4. What is the wavelength of the color of the car? (admittedly artificial)

All these are random variables on the same sample space.

Answer to question 1 might be useful to a person who sells eatables on the roadside? (more passengers means more business)

Answer to question 2 might help decide if it would be profitable to open a petrol-selling shop here.

I ask students to come up with examples of such statistical experiments instead of coin-tossing and dice-throwing ones.

I got this from a bright student:

Go to the library. Observe the first book that is borrowed by a user that day. Sample space is all books of the library.

Random variables are: Number of pages of that book, Price of that book, How many times it has been borrowed earlier.