Indeed, Slackware still defaults to LILO/eLILO and it works perfectly fine on modern hardware, older hardware from LILO's heyday should be no issue at all.
I still hang on to a PIII based Dell Latitude laptop from 2001 and Slackware -current runs surprisingly well on it (along with BeOS 5.1 from my original disc, QNX RTOS, OpenBSD, NetBSD, and a few other obscure OSes from its generation to today).
Yes, eLILO was made for this purpose. I still prefer it to GRUB, though EFISTUB is an option as well for those who don't want or need either, as long as your particular UEFI supports it.
Swapping hard drives, yes. Slackware boots on the original Windows 98 hard drive alongside Windows, and I have two other drives. One has BeOS 5.1, the other is my test drive I use to try out other OSes (NetBSD is on it right now if I remember correctly, but it hangs during boot sometimes). PATA laptop drives are getting scarce but I do have a PATA-SD adapter lying about that works in a few OSes and I have no shortage of SD cards due to my Raspberry Pi addiction.