Decide what you want to do first, and then build to it. You've got a widely scattered set of requirements.
...thought was I would reuse what I can from the 2013 into a new machine (case, power supply, maybe SDD, CD/usb).
That machine is a decade old. The only thing you might be able to reuse is the case, but they've gotten far better in the last decade with regards to silencing loud components. Power supplies do age out and fail, and your old PSU is unlikely to be suited to modern computing loads (especially if you have a decent GPU for ML).
Keep it as a spare, or sell it. Don't try to do anything with the parts in your new machine.
But then I got thinking it would be nice to have a laptop I can actually work on (coffee shop, park, travel etc). I could get a more powerful laptop but that comes at a cost was looking in the 1.5-2k range (did not look to hard or deep - did not look at refurb yet). But then maybe it would be worth getting a lower end laptop and doing the build with the intent to remote desktop to the desktop when I wanted to really use it remotely.
And this is how the money flushes out of your account. A "powerful laptop" is a horrible machine to use. I've had a variety for work over the years, and some for personal use. They're expensive, they're power hungry, they're hot, they're loud, and the battery life sucks. A "workstation laptop" device is a crappy desktop when docked, and a crappy laptop when portable.
Build a new desktop. Use it when you're home. If you must have a laptop, find an older (3-5 year old) lower powered machine, and accept that you use it for portable thinking and perhaps coding, but not anything intensive.
For a desktop build, I'd base around AMD, and spend the money on a decent size NVMe M.2 SSD - your decade old SSD is going to be glacial compared to a new NVMe unit. No need to go exotic high end WD Black or something unless you know you're going to thrash the disk.
My last, a screamer in 2016, alas is also capped at Win 10 because of the CPU & motherboard.
No, it's capped at Win10 because hardware OEMs were complaining to Microsoft that people weren't buying computers, because the decade old ones were still doing everything they needed. So Microsoft set an arbitrary CPU limit, not based on any actual features, granted themselves an exception for an older, crappy tablet they were still actively selling, and blustered about how it was important to have modern things. You could, last I looked, install Win11 on an "unsupported" machine with various warnings, and it would work fine, update fine, etc.
Why you'd bother, though, is beyond me, as Win11 is just an advertising delivery tool and behavioral data extraction tool with a side habit of running applications slowly. Put Linux on your machine when Win10 is EOL, keep using it, and don't look back. Or just ditch Windows now. It's no great loss.
Past ML projects have included deriving a neural net to play back gammon via reinforcement learning, this involved multi-day to week long runs. I have gotten moderately good at building GUI "apps" in ML professionally, current computer struggles to load some basic ones.
Ok, so build a powerful desktop.
Seriously, get it out of your head that you want to reuse old parts to save a buck. Build something that does what you want. Yes, you'll need a 750W PSU, 32 or 64GB RAM, 1-2TB NVMe drive. Build on a 5000 series AMD chip for some savings, and I've no idea what a modern GPU costs or what's required for ML stuff, but find something decent.
It's a hobby machine. Build it right now, and it'll last you a long time.
