I also fall into that camp. Network storage for something like TM sounds incredibly convenient and attractive. But in reality, solutions are either deprecated, slow, or finicky, or just plain unreliable. These days, I’m afraid it’s just a lot simpler to plug in a local drive.
Fortunately, thanks to inexpensive flash and sleek NVMe enclosures, these are small and light devices that can easily be plugged in on the fly. At home, for example, I tend to plug in an old MBA over night to juice up in the same spot. So instead of just plugging it in there, I got an inexpensive USB3 hub with USB-C PD support. The charger now goes into that along with a simple NVMe enclosure with some inexpensive flash. Now, when I plug that MB in to charge, it also gets to copy over all its hourly TM snapshots. Zero extra effort required.