Couldn't you say the exact same thing about the general election? They'd just do the campaigning beforehand instead.
The fact of the matter is that a staggered schedule makes some states more important than they have any reason to be; this is not good democracy.
Well, it certainly structures the Strategy Game that is a political campaign, like where and when candidates spend their time/funding. I'm not sure that it's fundamentally less democratic, though.
I'm not a sports person, so I hope not to botch this metaphore... but holding votes simultaneously is like determining the outcome of a basketball game by having one player from each team simultaneously throwing penalty shots into opposite hoops until one misses, whereas staggered ballots are more like the multi-person competition over one ball which teams normally play. In the second example, if a team is doing poorly, they can adjust their strategy to adapt to their opponents, decide what resources to commit to what function on their team, and adapt to changing conditions. In the first example, you can't strategize or adapt... you can only hope you sent your best player forward and prepared them well enough.
Whatever way it works out, you're ideally still seeing a competition to see who is the best player... but both of the games would be played radically differently. To leave the metaphore, simultaneous votes put a lot more focus more on a candidates travels, the presence of their message in mass media, and mostly require lots and lots of advanced preparation. Staggered votes put more focus on strategy and adaptation, and (I would say) create more opportunities to reveal the strengths and weaknesses of a candidate as they're put through a trial by fire.
If the concern is that primaries in each state influence those in subsequent states, a compromises could be keeping the primary ballots/caucus results secret until every one is decided... I think that would allow maximum time to campaign just prior to the decisions being made, without giving unfair influence to the first states.
EDIT: Still... that it affects later primaries isn't in question, but remember that each state still has equal chance and power to react to the primaries held before, and have equal say in the end process (per capita, of course). There's no true "advantage", so much as their is a chance to structure the playing field first, before other states get a chance to restructure it.