The first annoyance is that CloudFlare requires that you use their DNS servers, seems unnecessary to someone who isn't worried about being DDoS'ed, but okay, fine, I'll move one of my secondary domains (a .net) over to them.
I export my DNS Zone from Azure, try import it to CloudFlare and it can't understand the format since it's apparently not a proper BIND format. It's less than a dozen records so I just manually capture them, even though I find the UI for capturing DNS records clunkier than I would expect it to be.
Then I want to update my domain's NS records to point to CloudFlare's servers. My domain is currently an "App Service Domain" which is essentially Azure's DNS registrar offering (they're actually re-selling Wild West Domains services, which I think is GoDaddy) and it turns out it's not possible to update the NS records on Azure. At this point I figure the easiest thing to do is transfer the domain to CloudFlare as the registrar.
This is where CloudFlare has a total stuff up in their systems. Under the "Transfer Domains" section of their dashboard, it would only show "You currently have no domains available for transfer. Follow these instructions to initiate a transfer with the current registrar".
I look at the linked document, manage to get an auth code from Azure for a domain transfer. Still, the "Transfer Domains" screen shows the same thing. I check everything I can, I've captured the domain information on my CloudFlare account (showing a status of "Invalid nameservers", as expected), I check who.is and there is no indication that the domain is locked in any way, still, the "Transfer Domains" doesn't show my domain. I ask ChatGPT and it mentions it can sometimes take a few hours to show, 4 hours later it's still not showing.
I open a ticket and after a bit of back and forth they say the problem is that the "domain is not active", I tell them that to my knowledge everything is active with my domain and I ask them to tell me where I can see this status showing where the domain is "not active" and they tell me it's the status for the domain on the CloudFlare dashboard. Which (presumably) is due to my not having updated the NS records to point to CloudFlare, which I actually mentioned in an earlier email to them is not possible with Azure as the registrar, which is why I was trying to transfer my domain to CloudFlare!
In summary, it's impossible to onboard to CloudFlare if your domain is presently registered with Azure, their "smart" UI doesn't make it possible. I have had to transfer it to our Namecheap account which (as I would have expected on CloudFlare), simply allowed me to enter my domain name and the auth code on their "transfer your domain" page and now the transfer is in progress.
As a related aside, the reason I'm moving from Azure Front Door to CloudFlare is because despite a months long support ticket with Azure, they are not interested in solving the problem of cold cache downloads through their CDN being ridiculously slow, like 100MB/s (i.e. once the cache was no longer cold). The cold cache speed is less bad (but still not great) if you're doing a set up with everything in West Europe but we've had complaints from European customers in some countries of slow speeds even with West Europe as the source of the data, so I can only surmise that Azure Front Door is just generally terrible at serving files which are not yet cached.