As an approximation it's very roughly true, but it really depends on the parameters being used. It's not the center of mass for the solar system, that lies near but not in the Sun, just as the center, the barycenter, of Earth-Moon lies in Earth's Mantle. Generally orbits look more like this not as most people tend to imagine them. And the difference is not small or trivial from a practical point of view, you can't ignore it.

If you adjust your parameter as not being gravity, or just gravity, then you can arrange a definition in which the Sun is the center. But you can just as easily do that with lots of parameters. What is the center of Catholicism, or Fashion, or World Governance? And just like a binary star system, or a red dwarf with a large Jovian world, you can get an effectively empty spot as center. Like if someone points out that the Center of economics and banking is New York, or London, or Geneva, trying to find their effective middle, the 'true center' would just get you some empty spot in the middle of the Atlantic, where the only banking which might ever have taken place involved a pirate ship making a 'withdraw' of trade goods flowing from the Americas to London or Lisbon. And you really can't ignore gravitational perturbation caused by things like Jupiter anymore than you could ignore the effect of change in tax or tariff rates by a country on where the center of trade routes was.

In this case the 'right answer' isn't really right, and I wouldn't object to people calling the Sun the center of the Solar System if I thought most understood it really was just a generalized approximation and depends on how you mean it, the loose direction which 'down' generally points in the solar system. Like then center of a city being downtown, even though the 'center' is often nowhere near the physical midpoint of a map of a city, or even necessarily the center of a population density map. And if you need to get to City Hall, the political center of a city, the approximate center of the blob that is its city limits is as likely to be in the dumpster behind a deli a mile from City Hall as the mayor's office. Of course, everyone would naturally shrug if you pointed out and say 'you know what I mean' and of course we do, or 'the political center, man, don't be ass, same as X street's the center of the shopping district' but I don't think people realize that this is how the sun is the center. So when things like this come up, I cringe a bit at people laughing at the morons. Most fairly decently educated folk actually picture the solar system as anything different then a big yellow basketball with some ping-pong sized balls all within a few feet, even most scientists do because that's how we learn, and that's fine enough so long as one is at least intellectually aware that that model is terrifyingly wrong, and I doubt even 1 in 10 really do, whereas most people really do get that the 'center of Catholicism' is implying something very non-physical and not some spot many miles under the Earth's surface found by plotting the locations of every Catholic.

Incidentally the question is phrased right, since it only asked if the Earth was, not 'what is the center?', though one might fairly point out that the Earth is effectively the center of the Known Universe, since we can see equally far in each direction from it, or from a telescope on it's surface or in orbit anyway.