Can I simply attach a good portable back-up drive to an Xfinity "gateway' to have (reasonably) reliable data back-up?
No you cannot expect the vast majority of routers on the market to support Apple Time Machine.
Some can but they are usually high end wireless routers.. Asus or Synology for example.. Not gateway devices with inbuilt modem.
What I am referring to above is USB attached drive.. but you are are hoping for a simple solution to a much more complicated problem.. so you need some corrected thinking here.
(Am I correct in thinking a "gateway" is both a modem and a router, i.e., like Time Capsule, but without the back-up drive?)
You are correct that a gateway is term used for a combo modem router..
Where you need to revise that thinking is Time Capsule (TC) is a wireless router.. with internal hard disk.. it is not a gateway.. it has no modem.
The gateway does not have a USB port, so I would need to use an Ethernet cable between the gateway and the drive. This will require an Ethernet adapter, i.e., Ethernet connection from the gateway to an Ethernet to USB or USB-C adapter, depending on what the input to the drive is. Is the adapter apt make any difference?
You cannot plug an ethernet to USB adapter to the gateway.. so that is impossible.
I assumed from the first sentence that your Xfinity gateway had a USB port that supported hard disks. Since it doesn't you cannot add USB drives in any form.
You can however plug in a NAS.. That is Network Access Storage.. these are made by lots of companies.. the top two support Time Machine very well, Synology and QNAP.. but they are EXPENSIVE.
When you plug a NAS into the network.. the make and model of the router is pretty much irrelevant. The NAS is a device that has internal processor and network software with a hard disk.. it does not require drivers or help from computer or router.
So your second related question has the same answer.. you cannot do 1.. so you cannot do 2 either.
What back-up software do you recommend to replace Time Machine if it doesn't work with new equipment?
Carbon Copy Cloner is our favourite amongst the gurus here. But there are other popular programs.. eg SuperDuper and Chronosync. CCC IMHO offers the best combo of price, license conditions, speed and reliability.
But you cannot plug a hard disk to your router anyway.. it has to be a NAS.. certainly any device that supports Time Machine will support the others.. BUT beware the opposite is also true.. if TM is not supported.. neither can the other products use it.. Apple must backup files to a special format (Mac OS extended journaled) disk.. nothing else is supported.
I hope I'm not the only one who finds the topic confusing and potentially expensive.
Undoubtedly Apple in the past assumed you would buy a Time Capsule.. which was their version of wireless router plus hard disk for backup. It has now ceased production with no replacement. So Apple have surprisingly left people dangling.. with no viable network solution.
The cheapest and most reliable backup is to plug a USB drive directly to the computer.
For network backup.. then I would look at a cheap end NAS.. and strongly recommend Synology brand.
Some files you can store in cloud.. But Mac system files must be stored in correct format on directly connected disk or local network.
Bob has given you virtually the same answers.. and beat me to the punch!!
Certainly if you have a desktop Mac in your network.. it is possible to use a large USB external drive to backup laptops on the same network. This worked fairly poorly in the past.. but Apple has now changed with High Sierra to using SMB protocol (traditionally all Mac backup had to be AFP protocol).. and sharing a disk to the network will support Time Machine.. I have no idea of how well this works in reality.. But if you don't already have desktop Mac it is certainly no solution at all.. a Synology NAS is superior in most ways as a file server in your network.