Also some software expect to talk to a modem and don't play well with raw serial.
I searched for multi serial port card, but i had trouble finding one. Maybe i'm using the wrong keyword but i only found pci/pcie card with up to 4 serial port.
Usb seem to work decently but i admit that sometime the os (linux) seem to become confused and some serial port change name, causing some trouble. Having one card (or two) will make everything easier. I'm also considering "console server" as others have suggested.
As an alternative, PPP gives you a routed TCP/IP connection over a serial line...you offload routing onto the network layer and you can potentially have _all_ your client machines communicate with each other or even externally via standard TCP networking, and don't deal with circuit-switching the serial connections at all. For things that are TCP capable this is probably the cleanest solution.
If you want to support software that requires direct serial/modem links and not TCP, you will need something else in the mix. On the software side, I'd check the software to make sure you can't just turn off the AT commands altogether and use raw serial. Even where it wasn't a supported/intended setup, most software that explicitly handled modems allowed specifying alternate AT command strings, so you _might_ be able to just blank them out and go straight to serial. As far as circuit-switching...I know I've seen hardware devices that did this, but not in decades, so likely not easy to find any more. A quick google only turned up a bunch of datasheets for chips that could do it, but seems like you'd have to build your own device or track down a vintage device.
For multiport cards, an amazon search turned this up as the first result: https://www.amazon.com/Port-Rs232-Multiport-Serial-Card/dp/B...
So i, basically, do both. packet and circuit network according to need.
As far as circuit-switching...I know I've seen hardware devices that did this, but not in decades, so likely not easy to find any more
software based circuit switching seem to work fine for what i need, is really not that complicated, it also make me do cool thing like wiresharking!