I did professional fits for awhile in a shop. Your body's balance point relative to the crank centerline will determine how much weight is on the bars. Your upper body should be neutral with no weight on your hands while you are pedaling. The key factor controlling this weight balance point is your saddle setback. In a fit lab, sliding the saddle backwards is almost always necessary. You may even need a different seatpost with more setback to get the proper balanced position.
It is okay to have a very small amount of weight on the bars when you are stationary in the riding position, but it should still be quite easy to get into the riding position while leaning against a wall or with the bike on a trainer, then take your hands off the bars without feeling like you are falling forward at all.
I have a tiny bit of weight on my bars when stationary, but that goes away when I peddle. Like I can completely remove my hands from the bars while pedaling and stay in exactly the same position.
Once you have set up the saddle for a neutral position, only then is it time to find the right stem. Your stem should give you comfortable access the the lowest drop flats, and the main drop position as the primary factor, then the top positions are a secondary factor. Your physiology like your flexibility (and if you have a gut) will determine your lowest position you can sustain. The goal is to make sure all of the muscles in the torso that cross or constrict the lungs are disengaged. You can likely physically peddle at lower positions, but if you can't adequately breathe there is no point in trying to access these positions. If your lower bar positions are too low, you'll never use them.
Your arms should be a 90 degrees in the ideal fit position, if you are reaching, it will cause problems. A little short is okay but too short will make you hunch and compress your lungs.
The first step is always setting your saddle height for leg extension, then saddle setback for a neutral upper body.
