Bond duration and the mathematical proof of 'bond price recovery'

The term duration has a special meaning in the context of bonds. It is a measurement of how long, in years, it takes for the price of a bond to be repaid by its internal cash flows.

I have read this statement from the textbook and try to use the mathematical way to proof (the bolded statement) that is true. Thus, I have made up an example as follow:

Take the discount rate as 7% per annum

Term (yr)   Cash Flow   PV
1           100         93.45794393
2           100         87.34387283
3           1100        897.9276646

Fair value = 93.45794393 + 87.34387283 + 897.9276646 = 1078.729481
Duration = 1*93.45794393/1078.729481 + 2*87.34387283/1078.729481
3*897.9276646/1078.729481
= 2.745756684


Then I was getting stuck. When I try to add up the PV of cash flow at 2.7458 year, the result is not equal to the price of the bond (i.e. \$1078.729481)

Can anyone explain (in mathematical sense) why duration is a measure that calculates the time it takes for the price of a bond to be repaid by its internal cash flows , by using the above example? Rigorous proof by formula is also appreciated. Thans!