Doesn't a repeater only have to hear frames and then retransmit them? Why does it need the original Wi-Fi's connection details for itself? Can't it be a genuinely invisible middle man between the device and the original Wi-Fi network if all it does is rebroadcast RF information?
That's not all it does.
Originally repeaters did work this way (back in the era of WEP and 802.11b), but from what I've heard, they had various downsides – e.g. if I remember correctly, clients would see duplicate frames (reducing the channel's capacity), the repeater would indiscriminately repeat all networks on the same frequency, WPA couldn't be made to work (I can't remember why though), and I assume that if the same approach was applied to modern standards such as 802.11ac it would prevent new features such as beamforming or wider channels from working too as the AP would have no knowledge about where the clients are physically.
So modern "repeaters" are not layer-1 RF repeaters at all, they're layer-2 bridges that keep track of their own clients and overall act like additional access points in the network (except they have a wireless uplink connection to another AP, rather than wired Ethernet). It's just that the terminology has stuck – though it's not entirely inaccurate, as the device still repeats stuff, it's just happening at layer2 now instead of layer1.
(Wi-Fi natively supports roaming between multiple APs within a single SSID, so if your "repeater" is a fully functioning AP, clients will be able to handle it the same way as they handle multiple wired APs in range, and of course the repeater can handle clients on a different channel, it can do beamforming and stuff, etc.)
That being said, this method has its own downsides – e.g. a special "4-address" connection mode needs to be negotiated between the uplink AP and the "repeater" bridge in order to preserve real MAC addresses of bridged clients, which is something that mesh products do correctly but a RPi connected to a typical standalone AP will not be able to, leading to ugly hacks like L2NAT, but usually you won't notice those.