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I love the transformer, I wonder how they came up with that idea.
The Planar transformer[1] is a big upgrade from the traditional Wire-wound.

[1] https://en.wikipedia.org/wiki/Planar_transformer

Even wiki only list Advantages over conventional counterparts, what are the trade offs and drawbacks?

These days everyone is busy telling me all the benefits of something without telling me the price / cost / cons. I need facts so I could make up my mind. Not a marketing pieces.

the primary use here is because of its smaller size. the tradeoff is increased cost, complexity, and low overall power throughput related to size. which isn't really a problem for the RPi form factor -- but this wouldn't be indicated for higher current draw applications like radio, or audio i would suspect.
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I wonder why they don't integrate the transformer in the circuit board, that would be even more elegant.

Does the transformer PCB have more layers or heavier copper than the rest of the board?

the transformer does a lot of induction. you don't want that in your PCB; offset in a separate component is a better plan, especially considering this is a budget PCB and there's only so much mitigation they can do with the circuitry and mass production involved.
It would not be economical to do this unless the board is small. The PCB for the transformer is especially tuned for its performance: what dielectric is used, what copper weight, what distance between layers, how many layers, all these are complex parameters that need to be tightly controlled.

On the other hand, for the rest of the board, the PCB can easily get away with 4 layers on standard FR4 which is cheap to fabricate.

Using these low profile transformers is getting common, especially if you can't afford the height of a standard transformer on just one side of the PCB.

You should look into the magical stuff that powers small DC to DC power supplies: transformers entirely made of PCB, tight integration of the components inside the PCB itself sometimes. It's quite amazing really (and this is 5 years old, so imagine the progress made since):

https://www.youtube.com/watch?v=K6om51cr8Mk

Thanks for the video! Curiously, many of the boards in the video with planar transformers have the inductor integrated in the board rather than on a daughter board, so despite the difficulties it seems like it should be doable. Maybe for revision 3 of the POE hat.
In the smaller ones, the planar transformer takes up most of the board area, so there's no benefit to assembling two different PCBs.

The larger DC-DC converters are not as cost sensitive as a raspberry pi product; they tend to be used in low volume industrial/scientific/medical devices and might cost £70 each. When you consider that they are also mostly inductor by area, and factor in the additional complexity of manufacturing a multiple-pcb assembly, it's also not worth it.

IIRC there are also some larger PCBs in that video; for those it's possible that they already had enough layers for routing signals that it was easy to integrate the transformer core directly.

Another factor I haven't seen mentioned is the manufacturing complexity. It looks like the transformer could be placed by a normal pick and place machine, while assembling a transformer around a PCB would be more of a specialised or manual process. I suspect that might be the real reason, as the transformer PCB doesn't look like it has that many layers.

Step down converters are commonly used in car ECUs. Similar input and output range, power and everything goes on a single PCB. So there must be a different reason.
That is a power module made by Murata IIRC. I am jellying at that thick multilayer board.
Be advised that PoE splitters for rpi are also available from third parties for around $15.
And those are more usable for other devices with their general power jack (or USB) The Pi Hat only works for ... the Pi.
And also leaves the GPIO pins free for other uses
I have had three cheap ones in that price range fail. I suspect the Pi Hat version is better designed.
I am always wondering that there seems to be demand for a PoE hat. I have never heard of anyone using it. So who uses it?

Also how large is Raspberry Pi’s share in commercial applications?

I use one for a raspberry pi running a VPN, because it was a bit more convenient in terms of wiring (everything else on the network is also already PoE).
It's not a commercial application, but I have 2 raspberry pis rack-mounted[i] running some utilities for my home network (pi hole, gps-based NTP, ads-b data for FlightAware). Using PoE means no USB power bricks, no USB cables running all over the rack, and one less thing I need to plug into my UPS.

[i] https://www.amazon.com/dp/B08F9X528J

I've seen them used in an industrial setting, just to run a browser-based ERP client. Makes the wiring a little bit nicer.
I've seen some of these used for digital signage with PoE and network boot.

That way you can remotely power-cycle it and it grabs the OS and its config from the network at boot.

It makes the rollout of updated system images quite seamless.

It's great for racking Pis (e.g. my setup that I just posted about last week: https://www.youtube.com/watch?v=LcuNc4jz-iU), and also when you want to use a Pi in some location where running a power line would be cost prohibitive but running an Ethernet cable with PoE would be okay (for many, you have to have an electrician do the former, and likely get a permit).
I power my 4 node RPi cluster and my RPi based router via PoE.
>In general, we’re weathering the shortage very well, and the supply of mainline Raspberry Pi computers, Zeros and our other products have not been affected (we’re very good at pipelining).

This is a surprisingly false claim. I make a Pi-based product,[0] and there have been Pi shortages throughout 2021. Right now, I'm struggling to find any vendor that has Pi 4B 2GB or 4GB for anything more than one per customer.

[0] https://tinypilotkvm.com

Could it be primarily English shortage? At least one of my go-to store says “4+ months of stock” and couple others from the Foundation’s store finder shows them available. Not that I doubt the shortage but if you’re desperate finding some.
Oh, that could be. In which country are you seeing Pi 4s in stock?

All the vendors in North America are out, and Chinese suppliers on AliBaba are charging 2x the normal retail price.

??? Central Computers (NA with Chinese ties) seems to have some available right now (5/24/2021): https://www.centralcomputer.com/catalogsearch/result/?q=rasp...
Oh, thanks for the pointer. I've never heard of them, as they're not an approved reseller.

They're limiting orders to 20 per customer. They're selling the 2 GB Pi 4B boards for 42% over MSRP, but I might bite the bullet and pick some up.

Update: I tried purchasing 20 of the 2 GB and they told me they don't expect to have any in stock until October. They're still listing them as in stock on the website, though. And they say they actually can fulfill 4 GB orders immediately.
Germany, Switzerland, France, Singapore, etc., though the one I originally looked up was Akizuki as they're familiar to me. Distributors that have it in stock are either not shipping globally or have no English page, or both, so there might be a bit of difficulty and extra costs for arranging proxy shipments. Below is random pick from "Industry" tab in "Buy Raspberry Pi 4 Model B"[2] page.

- buyzero.de, $70, "Lieferfrist: Sofort versandfertig, Lieferfrist 1-2 Tage"

- pi-shop.ch, $67, "Verfügbarkeit: Sofort-Versand ab Lager"

- yadom.fr, $74, "Disponibilité : En stock", "commande passée avant 16H est expédiée le jour même."

- raspberrypi.dk/en/, $77, "_If_ the product is in stock: shipment within 0-2 weekdays.", "In stock"

- sg.cytron.io, $64, "Availability: 41", "Limit to 20 unit(s) per order."

1: https://akizukidenshi.com/catalog/g/gM-14778/

2: https://www.raspberrypi.org/products/raspberry-pi-4-model-b/...

Thanks for those pointers!

I commented on the blog post, and someone from Raspberry Pi actually reached out to me, so I might be able to buy directly from the foundation. But it's good to know about these alternative retailers in case I'm not able to buy direct.

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Sparkfun seems out of most too. I just placed a backorder for one.
POE without ethernet