OP - are you using a Weber chimney starter? It sounds like you aren't. The Weber chimney starter holds the coals above the ground and has vent holes in the bottom to allow the air to circulate - like the video you posted. You could always put the starter on top of the grate vs the ground. I put mine on a patio block or use the charcoal grate in the grill itself. The ash from the paper just falls through the bottom.
Putting the coals directly on the ground is most likely causing your issue. Lots of good ideas posted already about that.
As for grills, I use the
Weber Jumbo Joe as it has a built in lid holder to keep it closed while not in use.
Using a grill is going to give you a better cooking experience, but in some CG's the fire pit has a hinged grate that works goo for cooking over a wood fire.
Buy your Kingsford charcoal at Walmart. They have to 14 lb bags on sale all the time for 9 bucks. I get my cedar planks there too - great for cooking Salmon on the grill - over charcoal of course.