Some great discussion here so far, everyone.
As a relatively new DM (4 years of weekly 5e), the nonmagical weapon resistance was real problem for me. It started when I made a demon boss using the DMG rules, and as a low level monster with these resistances, it got shredded by my players. They all got a way to do magical weapon damage around level 5-6, and so all that power hidden behind b/p/s resistance was lost. I struggled to challenge my players by building encounters the way the rulebooks advised me.
Now I just max out the possible HP for enemies that are supposed to be any threat. My players are 19th level, and they have +3 weapons (when would they otherwise wield such artifacts?) and they are well optimized. They are not challenged by ancient dragons in their lairs. I have to do massive amounts of damage and double the HP of my bosses.
Magic weapons are just a toggle. There is nothing interesting about them. When I read about the struggles of older editions and their solutions, I can totally understand why the way things are as they are now. 5e wants to simplify things so players can spend more time rolling dice and less time doing calculations. With my high level players, they have all these bonuses and additional weapon dice and smite damage that it actually takes a little while to calculate all the damage. It is working like it is now, and the martials are almost overshadowing the spellcasters. So if you just play the game as it is now and use magic weapons, things sorta work out. Monsters do need a power boost. vincegetorix mentioned HP bloat, which is a numerical thing, but not a gameplay thing to me. There is a bigger problem with monsters not hitting hard enough compared to their CR, because all the "value" is pumped into their defenses, just so that they can get in a turn or two.
Do I want to calculate half of the damage more often? Maybe, it's not hard. Double it for vulnerabilities? Sure. Add bigger crit ranges, damage reductions or added dice? It would be more math, and I am not looking forward to it.
So the game as it is now works, sorta. Monsters are usually too weak as CR once intended it, even using Xanathar's rules. Resistance/immunity to nonmagic b/p/s damage will probably stay to show that some monsters can't be killed by farmers. It's more for flavour then, that these immunities and resistances are still there than for balance reasons. No cobbler will kill Orcus with a hammer. Also, DM's will have a mutiny on their hands if they don't hand out magical weapons soon enough. Published adventures tend to give magic weapons long before lvl5.
So what is the problem that we're trying to solve here? Do we want the game to become more complex and spend more time on what we already do now? Do we want flavor to have more effect on the rules? Do we want to make monsters stronger, as CR is a total shitshow?
In any case, I have learnt a lot about combat balance in my current campaign, which will finish up soon. I have a couple of fun campaign ideas lined up, and some of them might improve with additional rules for resistances/vulnerabilities to certain materials. If my players are interested in additional mechanics, I will look into them. I like the concepts of Leatherhead a lot, and I might expand on them.