I appreciate the in-depth response but I think you are misunderstanding the point of my comment.

My comment isn't denying that increased pressure at the source can increase flow rate (note I say that you can't increase pressure with a high flow shower head, I am not talking at all about changing the source pressure) - I'm just pointing out that people often conflate flow rate with pressure at the outlet, especially in domestic settings where the flow is intentionally restricted by design (like low-flow shower heads). In this case, pressure is maintained, but the flow rate is reduced by narrowing the outlet, which people mistakenly believe is a loss in pressure, while not realizing that they traded flow for that higher output velocity.

When I say that "flow and pressure are inversely proportional," I thought it was obvious that I was referring to how flow and pressure behave given a fixed source, since this entire conversation is about changing only the fixture. This is more about the relationship in practice when you change the outlet restriction. I'm talking about the "perceived drop in pressure" (what people mistakenly call pressure) when using a low-flow shower head, which is actually a result of lower water volume, not lower pressure per se. I'm definitely NOT talking about supply pressure and flow being inversely proportional, that's obviously not true.

So when Trump or others push for "high flow" heads thinking they'll get "higher pressure," they're misunderstanding how their own plumbing works. High-flow fixtures let more water through, sure, but if your supply system can't support that extra flow (especially with other fixtures in use), then the actual outlet force (again, what people call "pressure") feels weaker, not stronger. That's the irony I was trying to highlight.

Your garden hose analogy is solid, and I think you're mostly in agreement with my original point. You've just interpreted it as a misuse of Bernoulli, when I'm really commenting on how the misunderstanding comes from conflating pressure with flow, especially in domestic scenarios.

To clarify:

Increasing pressure, with no other changes, absolutely changes flow rate

I don't think I have disagreed with this - I mentioned replacing shower heads with higher flow shower heads, but of course that doesn't change the supply pressure, instead the loss in restriction lowers the velocity coming out of the shower head

Bernoulli’s principle that effectively states as flow increases, pressure decreases (and vice versa) when the source is not changed

I believe I stated the same when I said that they are inversely proportional when you are only changing the outlet nozzle

Since the bottleneck is more likely to be the miles of pipes and hundreds of bends, flow rate can’t really be inherently increased but pressure can

Yes, this is what happens when you put a low flow fixture, you trade flow for outlet velocity

The proper application of the principle is at the outlet with a steady source

This is exactly and solely what I am talking about

However, if you try to fill a bucket, you’ll likely find they fill at nearly the same rate

This is about the only thing I disagree on - you can fill up a bucket on the shower setting much faster than you can on mist, this is the entire principle behind low flow fixtures. If adding restriction to an outflow didn't reduce the flow then it would be pointless.

In short I really think we are mostly in agreement, I think you are mistaking my comment as talking about the relationship between flow and source pressure, which is definitely not what I am talking about since changing your shower head obviously doesn't change the characteristics of the water source. If there is something specific I said that doesn't agree with what you said then please point it out, because it really feels like you are just repeating my intended message.

It depends on how you make the showerhead / flow reducer. You could have one with a pressure reducer which will reduce pressure and flow and results in both less flow and force, but others just use smaller holes for a smaller outlet cross section, which trades flow for velocity. If the increased velocity is focused on the same or smaller area on your skin, then you'll feel an increased spot pressure compared to a lower velocity stream.

That's to say, plenty of low flow shower heads have focused streams, but they're generally too forceful and feel like needles. But there are ones in the middle that trade flow for velocity which makes it feel like the pressure is higher while using less water.