Correct. Unfortunately, the Mietpreisbremse is not as "automatic", because there is no fine tied to it. This means the landlord will not reduce your rent unless you do something about it. It also brings a lower reduction than the Mietendeckel.
Unfortunately, many rent contracts had a "shadow rent" clause - a higher rent they'd have to pay retroactively if the Mietpreisbremse was repealed. A lot of people are about to get a really big invoice they have to pay within 2 weeks.
Fortunately, the Mietpreisbremse is retroactive back to April 2020, so you can start the process now and get a refund later.
Your action plan should be the following:
1. Use a Mietpreisbremse calculator to check how much you are paying
2. Contact your landlord, either alone or with the help of a tenants' union (Mieterverein)
3. If that fails, use a lawyer, or a service like Wenigermiete. If it get this far, the resolution will generally take around a year. This is the average resolution time.
I have worked directly with Wenigermiete. They explained the situation to me just last week, but they expected the court to resolve the case in July, perhaps even later.