As an Americanist, there's 3 billion dozen cultures I wish people would talk about more, rather than being overshadowed by the "big three" Inca, Aztec, and Maya. But the public eye likes to latch onto the big easy things. Truth is, we don't know nearly as much about the Olmecs as we do their successors. The lives of the Aztecs were documented by Spanish and native scribes only 500 years ago. The same goes for the highland Maya. As for the lowland Maya of the Classic period (~pre 1000 AD), we have countless historical inscriptions and many towering, well repaired temples and other monuments. Olmec studies are much less accessible. There's not too many major sites, and they lack the instant "wow" factor of Tikal or Chichén Itzá. There's essentially only three "big" ones: La Venta, San Lorenzo, and Tres Zapotes. We haven't deciphered any texts (if they can even be callled Olmec). Perhaps the greatest difficulty is the Olmec's antiquity: La Venta dates to 1200 BC. Compare our knowlwdge of Mycenean Greece to ours of post-Roman Europe; that's a similar gap between Olmec and Maya.

I think we would be hard pressed to call the Olmec culture an empire. Did they have artistic influences on other cultures? Sure. But oftentimes people ignore what was happening within the region prior to Olmec "contact". Claiming the Olmec were an empire certainly takes away the agency of other cultural groups who may have chosen to adopt and use aspects and elements of the Olmec iconographic system for their own purposes, usually as ways of establishing and maintain power by having connections to other groups.

Other groups/sites during the same period include:

San José Mogote, Oaxaca

Paso de la Amada and Chiapa de Corzo, Chiapas

Tlatilco and Chalcatzingo, Central Mexico

El Opeño, Michoacan

Capacha, Colima

Nakbé and Cerros, Maya

Usulután, Southeastern Mesoamerica

Some of them lack strong Olmec influences El Opeño and Capacha or only take small elements like Tlatilco.