But saying grace with a meal, is "when there is bread present" So it's all about the bread.
Deut 8, in which we find the statement that we should bless (AFTER we eat, not before) only mentions bread in verse 3, where it says that man should understand that he does not live by bread alone, but by other things God provides, though even there it doesn't refer only to what we think of as bread. The word in question, lechem, sometimes refers to the baked item we call bread, but often, textually, refers to all foods (see repeatedly in Lev 21 and 22, speaking of fire offerings and sacrifices as "lechem") or is used idiomatically (as in Proverbs 31).
So, no, it is not all about the bread. It is about food. Rabbinically, there is discussion about what foods obligate one to say what blessings after them, and also, what constitutes "satiated", a minimum amount consumed in order to qualify for having to say a grace after eating.