Your understanding of the meaning is roughly correct. More precisely, the original sentence is short for
The Progressive “Active” formulas contain higher potencies of the key nutrients that are responsible for supporting a more active lifestyle.
The difference between “the (…) nutrients that are (…)” and “the (…) nutrients, which are (…)” is their scope. In “the X, which are Y”, the clause “which are Y” is additional information about all X. X stands on its own: you can shorten the sentence to remove “which are Y” without changing what the sentence applies to. In “the X that are Y”, the sentence only applies to the X that have the property Y. Concretely, the formula contains higher potencies of the key nutrients that support a more active lifestyle, not higher potencies of “the key nutrients” in general.
The original sentence is perfectly correct, and it's more idiomatic than the sentence with “that are”. In fact the sentence with “that are” is borderline because of the adjective “key”. Grammatically, “key” qualifies “nutrients”. But semantically, “key” qualifies “responsible for …”. The nutrients that the sentence is about are those which are key for supporting an active lifestyle. This makes it awkward to insert “that are” in the middle. If the sentence was just “… of the nutrients [that are] responsible for …” then adding “that are” would not make any difference.
It's effectively possible to use “responsible” as an adjective that appears after the noun. The adjective “responsible” on its own means reliable, trustworthy: “a responsible person” indicates a character trait, a pattern of behavior. On the other hand, “responsible for …” indicates that it's the person's job or task to do something. The “for …” part can be omitted if it's implied from context, and in that case “responsible” still appears after the noun.
The Progressive "Active" formulas support a more active lifestyle thanks to the higher potencies of the key nutrients responsible.