I've found the solution, and it's exactly as stupid and obvious as I was expecting.
The classroom computers were deployed using Clonezilla from an image that had the VirtualBox VM pre-configured. As a result of this, every VM had the same MAC address, which probably caused a lot of ARP collisions, since all the hosts and VMs were essentially on the same broadcast domain.
The solution was to simply randomize each VM's MAC address. After that, ICMP, SSH, and HTTP worked as expected. Thanks for the suggestions, but it was caused by my own oversight in the end.
(edit) I got around to reading the comments just now, @maxy@piefed.social was totally correct.
I know this isn't "selfhosting" as most people imagine it, but it is about hosting services on own hardware, hence why I'm posting in this community.
I'm supposed to help a teacher set up a networking exercise where pairs of computers are connected directly on a crossover cable and can access services (echo, HTTP, SSH, FTP) on each other. Every computer is identical: Windows 10 host, one VirtualBox VM running Linux Mint with a bridged adapter in promiscuous mode. Each host and VM has its own static link-local IP address.
The problem is, the VMs can't talk to each other, and I don't know why.
From one VM, I can ping itself, its host, and the remote host, but not the remote VM. Each host can ping itself, the local VM, the remote host, but not the remote VM. I've tried connecting both hosts to a layer-2 switch, with the same result.
Can someone point me at the one thing that I'm obviously doing wrong?
(edit) I've also tried to set the default gateway to the host's, remote host's, and remote VM's address, but nothing changed.

Running Linux on metal isn't an option. In the past, the classroom computers used to dual boot Windows and Ubuntu, but the Windows install got so bloated (the software too, not just Windows) that it needs the full SSD.
You mentioned that you can't ping the remote VM. Have you checked if the remote VMs IP is reachable? Is it in the ARP table? Check with
arp -aThe issue was ARP-related after all. Since all computers were cloned from the same image, the VMs ended up having the same MAC address, which caused collisions.
I checked
ip neighbour(it also shows the ARP table, so I assume they're identical), and it showed REACHABLE and STALE for addresses I could ping, but FAILED for the remote VM's address. I will checkarp -awhen I get the chance, though.