I think most people who switch to clipless take a few rides to adjust. It eventually becomes second nature. I myself have been using various systems for over 20 years now, and did indeed have a 'falling over' episode early on. It was a steep hill that I simply ran out of gas on. I fell over before I could unclip. Many thousands of km and 20-odd years later, and it hasn't happened since. Clipless is definitely the way to go, and imho there is no one 'best' system out there. They are all pretty darn good these days, with a few subtle differences that will take very little time to get used to.

I remember going MTB riding with friends back around 1987 or so. I was using clips on it, and my friends thought I was nuts for wanting my feet secured to the pedals. A few nutsacks colliding with crossbars convinced a few of them to try it. I went clipless shortly after and have never looked back.

For my city bikes or MTB, I use a MTB shoe that I can walk in and either Crank Bros. Quattros or Candys. For roadie-type or track use, I use Look Keo with non-walkable shoes. It's all good.