As I see more and more though it seems like the advantage to TEC is the cheap ships more than the strong econ
Very much so; the only areas in which TEC has a major economic advantage are with its Akkan Battlecruiser, Development Mandate, and Pervasive Economy. Other than these three phenomenal assets, the TEC economy is no better than the Vasari economy, and only marginally better than the Advent economy. Those higher-level civic technologies rarely come into play in a 1v1 scenario, and even in team games only appear later on, so in practice the Akkan is the TEC's only reliable economic edge.
Really the only reason I build them any more is for ice and volcano planets
That's the reality of a front-line player. You generally need a large and established empire to get bang for your buck from civic technologies, and this means you're usually 40-50 minutes into the game before they start to be desirable for a front-liner. Unless you're in a stalemate, however, by this point in the game you're usually knee deep in upkeep and you can't afford them anyways.
Should you really only invest in econ if you are in the econ slot?
No, you can and should do it on the front lines. You must be careful, however. You will always need to prioritize your military first. You will never become an economic powerhouse like a true eco player, but you can build a large and powerful empire.
I'm really close to starting every game with 2 military labs. People complain about rushing, but I really just don't feel sorry about it at all. Y'all let me know what you think since I haven't seen diplomacy or any 5v5 matchups.
Diplomacy is... controversial. You now need to research a technology to send money to allies. This prevents feed-rushes where one player is given an inordinate amount of cash by allies to beat his opponent, but it also makes it very difficult to survive an early 2v1 scenario. A 5v5 is one of the most brutal and rush-happy settings possible, but often times (due to the way teams usually end up) both sides will end up killing off one player and it effectively can become a 4v4 on a 5v5 map, which gives you the breathing room for some late-game economic tactics.