doesn't your explanation assume that all ac on a particular inbound route (when picked off for vectors) are following near as dammit the same track?
I tried to explain two scenarios-light traffic and when you are part of a sequence.
In light traffic (ie you are number 1) I think/hope it's self explanatory.
When you are part of a sequence and lets say the spacing required by the tower is 6 miles then you will always be 10 miles from touchdown as the preceding passes 4 dme.
If you are number 3, you will be 16 miles from touchdown when number 2 is at 10 miles
Now imagine you are number 4 and number 3 is on a straight in at 16dme, you have 22 miles from touchdown.
But if, for example, you are downwind just passing abeam the field and will eventually get an eight mile final, you still have 22 miles from touchdown but you are not going to follow the track of Number 3 (at least not until you turn finals).
By the same principal, no matter the tracks the aircraft take, you will always be 6 miles behind the preceding who will be 6 miles behind the one in front of them. It's just a question of knowing your 6 times table.
Another example - You could arrive at TIMBA at the same time as a company aircraft. If you are number 8 and he is number 9, you might have 46 miles from touchdown, then he has 52.
I could vector you across towards WILLO then give you a long downwind and long final to use up the 46 miles.
Your company aircraft could sit in the hold for 1 pattern (which uses up 20 miles) then get more of a standard circuit for the other 32 miles. You have not followed the same tracks but the track distances are correct.
Finally, we have to take into account the 'catch up' as aircraft slow from 220kts to 160. If you are going along the motorway at 70 and you come to a 50 limit, the distance to the car in front reduces as it slows down before you.
If you are 6 miles behind number 1, both at 220kts, he will slow earlier than you and the 6 mile gap will erode therefore we add a couple of extra miles to the track distance to take this into account. In the examples above I've disregarded the extra miles for the catch up in the interests of clarity !
I think I've managed to make a simple concept seem incredibly complex but I hope this helps. feel free to ask me to clarify anything (or pick holes in my explanation !)