There already are, but only for Europeans through the GDPR.
How it should be is that personal data's current and historical disposition is always available to the person in question.
If that's a problem for the company processing the data (other than being fiddly to implement at first), that sounds like the company is up to some shady shit that they want to keep quiet about.
Nothing to hide, nothing to fear should apply here, and companies should be fucking terrified with an existential dread of screwing up their data handling and looking for ways to always avoid handing PII at all costs. The analogy of PII being like radioactive material is a good one. You need excellent processes, excellent reasons to be doing it in the first place, you must show you can do it safely, securely and if you fuck up, you'd better hope your process documentation is top tier or you'll be in the dock. Or, better, you can decide that actually you can make do by handling the nuclear material only in some safer form like encapsulated vitrified blocks at least for most of your processes.
The data processing industry has repeatedly demonstrated they they cannot be trusted and so they should reap the whirlwind.
Fully agree on what you are saying, and my popcorn is ready for August when the penalties part of the AI Act comes into force. There is a grace period for two years for certain systems already on the market, but any new model introduced after August this year has to be compliant. AI Act+GDPR will be a great show to watch...