>Are we using ideas as an excuse to not start something?

Yes

>Should we randomly hack on things that we like?

Yes

>How did you get past your stage of analysis paralysis?

Start building something I love

>How do you pick what problem you'll be spending the next few years on?

Don't. It is far too hard to predict what is and isn't going to be a hit. If you really hate something, it'd be really dumb of you to pursue it just because you think you'll make money.

Build something you are passionate about.

I just keep a Trello list of any idea I have. I don't prejudge. I put ideas on there even if I think they're dumb because some day they might stop being or might morph into something else.