They have enforcement powers if they they prevail in the suit and the people who have legal power enforce their will. Which is something that has happened in the past. That includes them just successfully abusing the defendant into settling, no matter if they would have won in the long run.
The RIAA clearly has the power to enforce economic harm on anyone who has to defend their lawsuits. Against small enough defendants, that makes their positions extremely relevant.
Anyone can sue anyone. Just because they think something is illegal, doesn't mean it's actually illegal. They still need to convince a judge. Therefore their stance on whether you can rip CDs is irrelevant.
Lawsuits only get expensive when they can’t instantly be dismissed.
If I sue you for something ridiculous like using telepathic mind control to get my dog to bark satanic messages, the judge will just dismiss the case pre trial. If you launch a bunch of such frivolous lawsuits I can get a lawyer to counter sue you and win on contingency with zero out of pocket expenses.
The RIAA clearly has the power to enforce economic harm on anyone who has to defend their lawsuits. Against small enough defendants, that makes their positions extremely relevant.