I don't think networking is douchey. The example of rubbing shoulders and handing out business cards is an inefficient way to do it, but it does get results in some jobs [sales]. What you discussed with your client is the normal way. I think a different way is playing golf. A lot of executives play it, and it is a rich man's game. My dad plays, and he has met and befriended a lot of different known people, and even some politicians. Learning to golf and joining company hosted ones, or maybe even inviting coworkers to play can be a very good networking method.
This guy plays golf too! Unfortunately, I suck at golf and haven't played in thirty years :) It isn't a rich man's game here though. A lot of people play in the summer, and the season is "long" (only a few months, but since there is no night) and people play a lot.
There are lots of ways to connect without being douchey, but those networking events suck!
It isn't too late to go back and play. Golf isn't too taxing on the body, and can be good exercise. It is relatively cheap to play in the Philippines too compared to the US, but it still costs more than other sports. You need to pay to play [unless you are a clubhouse shareholder], you pay for the caddie [usually required], and the cart if you use one. You need golf clubs, bag, balls, and shoes. You need to pay for the driving range to at least know how to hit the ball before actually playing in a course.
Compared to sports like basketball or football where one only needs to buy is a ball, that is why golf can be considered a rich man's game.