Does this mean the processors have built-in USB and PCI controllers? Or is this more a case of "if you don't need many peripherals, you can do without a southbridge"?
I suspect the reason they can do without the southbridge is because many high-speed connections are PCIe these days (including storage and networking). But I don't see a good reason why Zen would have on-chip USB or RS232 controllers.
Socket AM5 has an integrated x16 PCIe bus, three x4 PCIe buses (one usually used for the chipset, one for NVMe, and one for USB4 ports - but other assignments are of course possible), a dedicated DisplayPort connection, a three-port hybrid USB3/DisplayPort interface, a dedicated USB3 connection, SMBus & I3C, and a HDA/SoundWire connection.
The AM5 Southbridge is essentially just a PCIe hub combined with some PCIe-to-USB and PCIe-to-SATA interfaces. If you don't need those, you can just leave it out! In practice you'll virtually always have it for the SATA ports and extra PCIe lanes, though.
Don't forget that USB is also a high-speed connection these days. 10Gbps is very common, and USB4 goes up to 40Gbps in Gen3 - with 80Gbps for Gen4 on the way. RS232 is indeed not included, but that's pretty much a dead port anyways: no reason to use it for onboard peripherals, and consumer motherboards rarely have a header for it these days.
AM5, I don't know.