I regularly fish three rods and occasionally fish four. I have the appropriate two licences to enable me to do so legally. For many years, I wouldn't have used more than two rods where others would use more. It took quite a while for me to get comfortable using three on an appropriate venue. I'm still not entirely comfortable using four. Waters quiet enough to fish four rods safely, and with enough space to not impinge on other anglers, are fairly rare.
However, I've never dreamed of fishing more than two rods on a river, and, more often than not, I fish one rod. Unless the riparian owner states otherwise, it's perfectly legal to fish three or four rods on a river, provided you have two rod licences. Everything depends on the water concerned, the methods employed and species targeted. As far as I can recall, I have never had a double hook up. Rather than luck (or bad luck, depending on you point of view), I put that down to me regularly having made the correct choices in what might be appropriate.
I'm not particularly concerned by this licencing proposal. Those that are sensible will still fish responsibly. Those that aren't sensible now, won't be any more or less responsible. If riparian owners believe that more than two rods is inappropriate for their water, they can easily make an ammedment to their rules as they see fit. Just because the law allows the use of two, three or four rods, doesn't mean that local rules can't be applied. I dare say that a minority will cheerfully ignore such rules. I'll bet that plenty of them cheerfully ignore the current rules anyway. I've long been less than convinced that the majority fishing three or four rods actually have licences for such anyway. Very occasionally, I've seen people fishing more than four rods, there's no way that they can be properly licenced for that, it's simply against the law.
https://www.gov.uk/fishing-licences