That is true. One way of recovering those funds would be to adjust the third bracket also, in addition to the second, that is to say increase it in order to recover a little money from that tax cut, if you really want to target the middle class.
You propose a reduction in the first tax bracket. But then you still have the same problem, which is that if you reduce it, a good part of the tax cut will benefit everyone. Once again, it's a matter of targeting and objectives.