Transfer Aperture library to new computer.

In terms of copying the library from the old machine to the new, would this be done by physically connecting them, say by a Thunderbolt 3 to USB-c connector, and dragging from one to the other?

Exactly

Honestly, I have significant reservations about Photos as a DAM, especially in the case of disaster recovery. I suggested CaptureOne as you mention it. Mylio is good too.

First off: transferring the library to a new machine is straightforward - just copy it from the Pictures Folder on the old Machine to the same location on the new machine. If you have a reference library then consolidate it first.

As for migrating, that really depends on a: where you're going to and b: what exactly you want as your end point.

In the case of a: some apps have a facility to import/migrate an Aperture Library (often with some limitations). Lightroom Classic, for instance has such a facility. There is a route from Aperture to CaptureOne as detailed here:

https://learn.captureone.com/blog-posts/migrating-apple-aperture-libraries-to-capture-one-pro/

and it's well worth googling around for more.

In the case of b: it's important to be realistic. Aperture preserved your Master and edits in a relationship that makes re-editing simple, and non-destructive. So do CaptureOne and Lightroom Classic - but all three apps manage that in a different way, so you may need to choose between migrating only your originals or migrating your originals plus edits, but without a relationship between them. So that's a thing to check out in whatever app you choose as a replacement.

Photos for Mac can create a Photos Library from your Aperture Library and save the albums and many of the metadata. It will keep the originals and the edited versions paired, so you can revert the adjustments. I have converted all my Aperture libraries to Photos Libraries, to give me access to may Photos on macOS 10.15 Catalina and macOS 11 Big Sur.

Whichever application you pick as a replacement, none of them will be able to recreate the Aperture edits completely, afaik. To save your Aperture edits as they are I recommend to create high resolution previews in Aperture, before you migrate the library to the application of your choice. Some Aperture users prefer to save storage by using only small previews of the edited versions. The migration to Photos requires definitely high resolution previews.

If you have been a power user of Aperture, you will find Photos for Mac limited. But it offers a quick and lossless migration. The originals and edited versions will remain paired, you will save the albums and most metadata. I like it as a means to get access to my Aperture Libraries and have migrated my libraries from Aperture to Photos as a fall-back, to be able to open them on a system that does no longer can open Aperture. You may want to create a Photos version of your Aperture Libraries, while you can still run Aperture. Just as an Apple supported fall-back option. Create Photos Libraries from your Aperture Libraries, just in case, while Aperture is still supported on your Mac. But check out the other applications recommended by TD, if they suit your needs. I like Photos in spite of its limitations because of the iCloud Photos integration.

I can use the same iCloud Photos library on all my Macs with Mojave, Catalina, Big Sur and my iPhones and iPad as well. But I have to supplement it with Apple Scripts and HoudahGeo for batch changing metadata and to use third-party editors to supplement the limited editing capabilities. The built-in editing tools in Photos are covering 80% of my needs, and I am mostly using GraphicConverter, Affinity Photos, DxO Perspective, Aurora HDR 2019 and Luminar 4 for advanced editing, mostly brushed in adjustments, lens correction, perspective correction.