Halal/Haram in Quran+Hadith is not 100% the same with halal certification/lack of halal certification in reality.
Also note that the lack of halal certification is not directly equivalent to something with a 'haram certification' (probably don't exist).
First let us agree that we should only eat Halal (Quran) food and refrain from Haram (Quran) food.
Should we eat halal certified food? With high confidence we can, since they are checked strictly by Muslim food supervision companies as you mentioned. Note that this high confidence is not 100%, because humans make mistakes.
Should we eat non-halal certified food?
If the lack of halal certification comes from the fact that the food failed the certification test, then with high confidence we should not eat this with same reason
if the lack of halal certification comes from the fact that it did not undergo such a verification process, then we are left to judge with our best judgement according to Quran.
But how do we know which case it is? We don't, but we can approximate. If you are in Saudi Arabia for example, and you see something not marked halal, then it is very probable that it is the first case where it failed the certification process. If you are in China for example, where most goods do not go through such verification process, then it is likely to fall in the second case.
Therefore in the end, you still need your own judgement to the best of your knowledge.
This is just my reasoning that I toss here. And Allah knows the best.