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In the last decade, people have been saying that high pressure tires don’t actually make you faster. My takeaway from what Jan Heine says on Bicycles Quarterly: either there is suspension loss in the tire or in your body, so there’s suspension loss either way: you’re no faster or slower.
So why do I feel like I roll faster with less effort when I pump up my 47mm shallow-tread city tires from 3 to 4.5 bar?
I think you are comparing apples to oranges.
You have to note that most people who ride on low rolling resistance tires do so on ridiculous 23mm tires because they do so on ridiculous bikes that won't take any wider tire (and won't take any mudguard for instance). They have to pump it up to very high pressures in order to not pinch flat. The pressures are so high that the poor suspension in fact can create losses. We aren't talking about 3 bar or 4.5 bar, we are talking about over 8.5 bar.
If going to the territory of reasonable bikes, you can fit wider tires (and you should since wider tires roll better -- they have minor air resistance drawbacks at high speeds though). In fact, low rolling resistance tires such as Continental GP 5000 are today available as wide as 32mm, which you might pump to about 6 bar.
Your 3 bar is so low pressure that it obviously is excessively low, thus allowing reducing rolling resistance by pumping the tire up more.
And I bet no 47mm tire with tread pattern is going to roll well. Good tires are available no wider than 32mm, and these have practically no tread pattern. Oops, you can't sell a tire with no tread pattern since everyone knows tires with no tread can hydroplane (well perhaps on a car but not on a bike). So they fit a micro tread pattern to assure riders it won't hydroplane, a truly unnecessary feature. Fortunately the micro tread pattern is so minimal it's practically no tread pattern.
The important thing to realize is, that there is always an optimal tire pressure for each combination of tire, weight and road bumpiness. If the pressure is too low, you will loose too much energy massaging the rubber of your tire, and if the pressure is too high, you will loose too much energy by the tire bouncing off of the bumps in the road surface. The optimum is where tire is squeezed as little as possible by your weight while flexing away the vast majority of bumps so you don't feel them in your hands.
As such, optimum pressure depends heavily on road surface. If you ride on a washed-out compacted sand path with stones sticking out everywhere, you probably need a pressure somewhere between 2 and 4 bars. If you ride a mountainbike trail, pressures below 2 bars look good. However, if you ride on a freshly built road, your optimal pressure is well above 5 bars. For such smooth rides, the pressure limit of your tires is likely lower than the optimal pressure.
As such, if you compare the feel of your different pressure on smooth tarmac, higher pressure will almost always be faster. However, most riding is not done on perfect roads. Some roads will have old tarmac, some roads may have potholes, some routes will have compacted sand (in different states of quality) or even plaster. The typical bike needs to handle all of these well. And on these surfaces, higher pressure does not feel faster at all, it actually tends to feel significantly slower.
