It would be like if I went into H&R Block to get my taxes done. Should I be worried that they have my SSN and earnings? No, because they need that information to do the task that I'm asking of them. If they sell that to other places yes, that's a huge issue.
So a privacy risk is the cost of doing business. Knowing about it upfront and consenting to the data collection doesn't negate that. If H&R block stores all of the data you give them forever, that's a privacy concern because they could be hacked or they could (depending on relevant laws) change their mind and decide to share or sell some of that data in the future.
You're arguing that anyone knowing anything is a privacy concern which is over the top in my eyes. You could never use a bank, file taxes, get a paycheck, hell you couldn't even ever get a job with it being a "risk" because your employer has a ton of info on you.
Being male I'm not familiar with the features offered by period trackers, but is there any valid reason the information has to leave your device at all?