I pay someone to make a 3rd party reduce their emissions so that I can fill that gap again.

If you do that, sure, then we have a problem. But so do you, since you may end up in jail. If you call a product carbon neutral in your advertising, you'll need to have a reason to believe that all emissions from that product were offset, otherwise you're guilty of fraud. So - in the context of that product - there's no legal way to start emissions again. Obviously in practice there's a lot of wiggle room regarding what emissions can be attributed to your product and how well the offset works and you'll have in dubio pro reo on your side, but in principle offsetting actually erases the entire carbon footprint of a product.

We're not speaking about emission rights here. In those cases, yes, shifting them from one entity to another doesn't directly decrease emissions but it still helps since it makes emissions more expensive and therefore leads to companies looking for ways to avoid them.

Edit: The 3rd parties in case of emission offsetting don't tend to be limited by emission rights. We're talking about thing like giving solar cookers to farmers in the developing word. That example these people an option to avoid using coal and helps with the climate, but it also helps them avoid the health problems that come with coal fires.