Does bread flour (vs AP flour) make a big difference in the density of the dough?
Bread flour has a higher protein content than AP flour.  That should aid in the development of the gluten structure of your dough.  So, bread flour will help, but that might not be your only issue.  No-knead, doesn't necessarily mean mix it together and leave it alone. (I did watch the video...I see he doesn't knead, but his dough does not look like it is a high hydration dough). There are many variables that influence water absorption in flours...so often a formula will behave differently in your kitchen and with your ingredients.  From your description it sounds like your dough needs more structure, and would benefit from the use of techniques that build that structure.  The most obvious is kneading, but slap and fold (or stretch and fold) is often incorporated into high hydration formulas.  There is a learning curve, so reducing your hydration is a good way to get a feel for things...but I suspect that you need to work on building the gluten structure.