There is a concept of eid echad ne’eman bi’issurim, that is why we are allowed to eat anything that we haven’t made all the ingredients ourselves. When it comes to eating in other people’s houses this is what everyone relies on. But, if the person tells you themselves that they rely on certain kulos than you have to know what you personally hold regarding those kulos and how it affects their kailim regarding you. For this, you have to ask your own Rav.
When it comes to food items that are sold, most people nowadays are machmir not to rely on the seller as an “eid echad” because he is nogea b’davar, that is why we have mashgichim and hechsherim.
This is all regarding items where we know what they are, what the ingredients are, etc. But, most people nowadays don’t know (and usually have no way of knowing) what goes on in every step of the process with commercially available foods. For this reason, we need hechsherim. Even if you know the ingredients, you don’t know about the machines used to put the ingredients together, etc. so you need a hechsher. This is the mainstream view of Orthodox Jews. There is room to be maikil in certain cases but in places like NY/NJ where there is plenty of food available with hechsherim, most people do not rely on these kulos. Rabbi Abadi obviously holds differently.