League of Shopkeepers worked as it should in the game.
It gave you the 5 Favor reward (+$1) and later the additional 10 Favor reward (+1 Action, +1 Buy) after you played the Liaison (Broker). That happened during the Action phase (as Broker is an Action card).
However, with the exception of turn 11, where you could play a Kiln afterwards, you never had other Action cards in hand. That means that the game advanced immediately to the buy phase. If you had had an Action card in your hand (as in turn 11), you could have taken advantage of the +1 Action from Broker, once you had 10+ Favors.
You are generally right that the +1 Action from League of Shopkeeprs is useless if you play the Liaison during your buy phase (e.g. if it is Bauble or Contract). When cards are designed, not everything can be beneficial all of the time. That's especially true for the Favor mechanic, that is distributed among 2 kinds of cards (Liaisons and Allies), where some combinations are not as good as others.