No, the GOP has a small natural advantage. Winning a state by a few percentage points, especially if you're, like Bush, an incumbent running in good times against an unattractive challenger, does not mean the state can reliably be placed in your (our) party's column.
However, incumbent senators, or at least incumbent Democratic senators, have a large natural advantage.
Just because Democrat Senate candidates can win in Republican states does not mean that the GOP does not have a huge natural advantage. If Senate seats were apportioned by population (so California would get 53 and Wyoming 1) and all Senate seats were elected statewide (as they are in the actual Senate), then neither party would have a natural advantage, since Republicans and Democrats are fairly evenly divided in the Electoral College, especially after subtracting 2 EVs from each state. However, since each state elects 2 Senators, and there are a lot more states where Bush won by over 10% (22 of them) than states where Kerry won by over 10% (7 of them), yes, there is a huge natural advantage for the GOP in Senate races. There just happen to be a lot more small states that vote heavily Republican than small states that vote heavily Democrat---16 states with fewer than ten electoral votes gave Bush a 10%+ margin in 2004, while only 3 states with less than ten EVs gave Kerry a 10%+ margin---and thus the equal representation in the Senate helps the GOP.