No, Commonwealth doesn't "make economy irrelevant to stability". It simply eliminates stabliity penalties for low numbers of traded resources and an economy in deficit.
These are each just minor components of the economy stability factor, which is actually measuring quite a number of things. Overall, it is a measure of how much your economy is growing. If you have 1-star rating, then your economy has been stagnant or maybe even going backwards for some time. I almost always get a 5-star stability factor for economy. It is one of the easiest to get up to 5-stars in, compared to foreign for instance which is much harder to improve.
To make your economy grow, you need to increase the amount of commerce your cities bring in from the last turn. This can be done by building cottages, which then grow to hamlets, villages and towns and thus continually grow your economy. Replacing some mines with windmills is another way, as is increasing the value of your trade routes by diplomacy or by building lighthouses/custom houses.