Just tried with the same Waveshare HAT you had pictured above:
It still would not show up in lspci:
$ lspci
0000:00:00.0 PCI bridge: Broadcom Inc. and subsidiaries BCM2712 PCIe Bridge (rev 21)
0000:01:00.0 PCI bridge: ASMedia Technology Inc. ASM2806 4-Port PCIe x2 Gen3 Packet Switch (rev 01)
0000:02:00.0 PCI bridge: ASMedia Technology Inc. ASM2806 4-Port PCIe x2 Gen3 Packet Switch (rev 01)
0000:02:02.0 PCI bridge: ASMedia Technology Inc. ASM2806 4-Port PCIe x2 Gen3 Packet Switch (rev 01)
0000:02:06.0 PCI bridge: ASMedia Technology Inc. ASM2806 4-Port PCIe x2 Gen3 Packet Switch (rev 01)
0000:02:0e.0 PCI bridge: ASMedia Technology Inc. ASM2806 4-Port PCIe x2 Gen3 Packet Switch (rev 01)
0000:03:00.0 Non-Volatile memory controller: SK hynix Gold P31/PC711 NVMe Solid State Drive
0000:05:00.0 Ethernet controller: Realtek Semiconductor Co., Ltd. RTL8125 2.5GbE Controller (rev 05)
0000:06:00.0 Ethernet controller: Realtek Semiconductor Co., Ltd. RTL8125 2.5GbE Controller (rev 05)
0001:00:00.0 PCI bridge: Broadcom Inc. and subsidiaries BCM2712 PCIe Bridge (rev 21)
0001:01:00.0 Ethernet controller: Raspberry Pi Ltd RP1 PCIe 2.0 South Bridge
Is it possible it’s a power supply option? I’ve tried two different Raspberry Pis, and four different HATs now, so there’s something definitely happening that’s throwing me for a loop, haha! Maybe I’m in the middle of some strange singularity in space that’s causing PCIe signaling/power to fail.