I played more of "Secret of the Old Clock". It turns out that you don't have to solve Topham's logic puzzle, in order to search his house. You have to solve the puzzle, so you can talk to him. Well, you don't need to talk to him, until you need the quartz for the ham radio. So you can totally put off solving the logic puzzle for a while.
To trigger the trivet fetch quest, it looks like you can check the notebook at Topham's OR you can read the typewriter ribbon at Archer's. It takes about 13 seconds to get it at the notebook, and about 30 seconds to get the typewriter ribbon. That's mainly because Nancy reads every part of the note off the ribbon, where she just comments on the note off the notebook. And you have to open the notebook to trigger "ask Emily about her mother's middle name", anyway, so let's go with checking the notebook as the faster way to start the trivet quest.
I also learned you don't have to play the golf flag game! If you know the answer ahead of time, you can just input it into the clock in the carriage house.
When I finished my first trip into town, Nancy's dad called, and I could find the hidden passageway in the den. So maybe "get the mirror from Jim Archer" is the trigger for this. Or maybe "make a trip to town and back" is the trigger. Either way, I can put off solving the bird clock puzzle, until after the first trip to town. That way, I can get the mirror in the den at the same time I find the hidden passageway. I have to go into the passageway to get a dollar. The next step is testing the triggers for the Creepy's Corner puzzle in the passageway, to see if any can be skipped.