Upgrading High Sierra to Catalina on 2016 MBP using local webserver
In the past colleagues (and I) believed it was often a bad idea to install newer OSs on older Macs.

It's actually a bad idea not to upgrade to at least more current versions of OSes as you will miss out on important security and feature updates.  There's no evidence that Mac's can't or shouldn't be upgraded.

What I'm wondering about is whether my local web server will survive the upgrade.

It depends.  When Apple releases a new OS, they try to update all of the 3rd party software as well. This means if Apache changed things from one version to the next, you could potentially break something. Without knowing which version you had, which version you're going to, what your config looks like specifically , it's impossible to tell you whether it will break or not.  However, in the interest of playing it safe, assume it will.
The ideal path is to keep your web server up and running, install/upgrade your OS onto a different machine, deploy your web server, test it out and once everything is working switch over.  It's never a good idea to take a running (production) web server and just upgrade the OS in place and hope for the best.  It's not a good idea to do this with individual services for that matter (i.e. Apache, MySQL, PHP, etc.)

Any of you experienced migrators have an opinion about which way I should go?

Use Virtual Machines
Unless you have a special use case, I advice folks to not load up an AMP stack on macOS.  Why?  If your site is being hosted somewhere it's likely a Linux variant or FreeBSD.  You want to keep your test/dev environment as close to your production environment as possible.  When you have a VM, you can easily manage the web server environment independent of the host OS.  You can also clone the VM, perform the upgrade as I described earlier and if it succeeds, simply forward traffic from the firewall to the new VM - you wouldn't have to purchase a single piece of hardware.
In my use case, I standardize on FreeBSD and use only the bare minimum that I will need for my AMP stack.  This keeps the overhead for the VM as low as possible. There's no desktop environment, no client software, nothing but the core OS and the AMP stack with necessary dependencies.  I usually have my DocumentRoot pointed to an iSCSI LUN on a NAS or SAN (locally or in the cloud).  This way, I can very easily move data from VM to VM without much downtime.  My NAS has it's own multi-level backup so I can be certain data is secure.
These are broad stroke suggestions and you'll need to tailor it to your  specific environment.  By using VM's to host your web server, you gain a lot of flexibility especially if you want to migrate from a locally hosted machine to a cloud based instance.  It's very easy to copy a VMDK to an Azure or AWS container and then scale as needed.  Since you're considering a migration/upgrade, now is the time to do this - migrate to a virtualized environment.