Past participle as adj

I won't comment on rigged as a "be-verb" because that term means absolutely nothing to me.

However, it's the and which is causing the problem, and my intuitive response is that your boss is right. The and means that rigged is to be considered with driven. Using brackets to define components of the sentence, it's

The system is [rigged and driven] [by greed].

Here, rigged and driven are both past participles functioning as adjectives. Both are describing "the system". Because they are both exactly the same part of speech, with the same qualities, joining them together with and means that they are considered together, as a single unit — as shown by the bracketed sentence above.

If the sentence had been something like "The clock is blue and driven by a motor" then the motor would have nothing to do with blue because blue and driven are different.

To separate rigged from driven by greed, simply remove the and. You do need to introduce semi-colons, though, because it's essentially a list of what cynics tell you. Making it a list also allows commas within the list items.

Cynics will tell you that everyone is selfish and weak; that the system is rigged, driven by greed; that you can never succeed, so it's pointless and contemptible to try; that all ideals are ridiculous, and the do-gooders are only out to show off their own supposed virtues.

The last comma might be a semi-colon, but in that case you would need "...ridiculous; and that the do-gooders..." because the that is being repeated in each list item.

rigged is a part participle. Also, https://www.merriam-webster.com/dictionary/rigged suggests that we may treat rigged as an adjective. This being the case, the sentence is correct from two slightly different viewpoints. It is up to you and your boss to agree which fits the circumstances.

In the first (participle usage), the system is rigged by greed and the system is driven by greed.

In the second (adjective usage), the rigged system is driven by greed (it may have been rigged by some other motive than greed).

I'm not sure what your boss means by be-verb. Perhaps he's thinking of languages like French or German, where some intransitive verbs form the compound past with a form of to be, rather than a form of to have. At any rate, I think that there are two ways to interpret the sentence. The most natural to me would be to consider "rigged" and "driven" as adjectives. But I think it would also be possible to say that the sentence is in the passive voice: someone rigs and drives the system. This is reinforced by the "by greed" clause, since in the passive voice, "by" is used to introduce the agent. Since the actual meaning is the same in both cases, however, I don't think it makes a difference. What is clear is that it isn't the system which is doing the rigging or the driving.