Does a family really keep addiction going? Asking therapists and people in recovery

I work with families of people addicted to alcohol and drugs in Ukraine, and there is one idea I keep running into resistance about, so I want a sanity check from people outside my bubble.

The idea: the family cannot cause addiction and cannot cure it, but it can feed it. Paying the debts, calling the boss to say "he's sick", cleaning up the apartment after a binge, giving "one last" loan. Every act of rescue removes a consequence, and consequences are the only thing that ever pushes an addicted adult toward change. Families hate hearing this, because it sounds like blame. It is not blame. It is leverage they did not know they had.

My questions:

1. If you are in recovery: did your family's rescuing keep you using longer? Or is that a myth we counselors tell mothers to make them stop crying?
2. If you are a parent or spouse: what was the moment you stopped rescuing, and what happened after?
3. Therapists: do you frame it as "you influence but you are not guilty", or do you avoid the word "influence" entirely?

Background on how we work with this, if useful. We run a long-term therapeutic community (a year, not weeks, no locks, no "coding" or hypnosis), and families do a parallel free course. I described the family side here: Edit [Moderator]: Link removed and the program side here:
Edit [Moderator]: Link removed (Ukrainian and Russian, browser translation works).

Not looking for clients, this is an EN forum and we are in Ukraine. Looking for honest experience.