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Set is specially overloaded for Part in order to allow in-place modification of the expressions:

a = x -> 2;
a[[1]] = 1; a

1 -> 2

It is documented on the Documentation page for Part (the first point under "Details" section):

You can make an assignment like t[[spec]]=value to modify any part or sequence of parts in an expression. »

For your particular purpose the simplest approach is to replace Rule by Set:

Clear[x, y, z]
{x -> 2, y -> 5, z -> x + y} /. Rule -> Set

{2, 5, 7}

Another method is to Apply Set effectively replacing the head Rule by Set:

Clear[x, y, z]
Set @@@ {x -> 2, y -> 5, z -> x + y}

{2, 5, 7}

Inside of Do:

Clear[x, y, z]
Do[Set @@ a, {a, {x -> 2, y -> 5, z -> x + y}}];
{x, y, z}

{2, 5, 7}