Gavin, first, let me caveat that what I'm about to say is not legal advice; any questions around GDPR should be run by proper legal counsel in order to ensure your company is completely in compliance.
The "email this article" function may have bigger implications than just GDPR compliance and unsubscribe links. It means anyone can put in anyone else's email address and send them an email. This can mean big deliverability issues for you if you're sending through your own IP address—if you get hit by a bot or people input a lot of bad email addresses, that can at best dirty your IP reputation, and at worst put you on a blacklist.
Has your team considered using a "mail-to" link as a simpler option? In general, sites that offer the option to email an article or link will do so by using a mail-to link that requires the requestor to send through their personal email client, rather than through an in-app modal. Since you're not saving any of the email data anyway, could that be a better solution?
Gavin, can you give some more details around what the purpose of this email is? If your email is required for the person in question to access your site, their account, etc. (like a password reset or an update about their account), then it can be termed operational and wouldn't necessarily need an unsubscribe link. However, if it's not operational, it definitely needs an unsubscribe link in order to be compliant with all countries' spam laws.
Hi Matthew, I think the important question is not what the email client's market share is overall, but what percentage of your audience is viewing your emails on that client? Your brand's audience will have different behavior from anyone else's, and knowing which clients they use the most will help you focus your time and resources when building.
Tools like Litmus Email Analytics can show you which devices your subscribers use the most to open your emails. You can then make a more informed decision about which clients you need to spend time making sure they look 100% correct—and which ones have such minor share of your audience that you don't need to worry about them.
If they don't take action to opt in via the re-permission campaign, they reside in the EU, and you don't have an explicit opt-in on file, you aren't supposed to send marketing emails after May 25th. I wouldn't delete them unless they write in asking you to do you—instead, create a suppression list for those who don't confirm their subscription.
In my experience, an unsubscribe click doesn't count toward the click-through rate, for the very reason that you pointed out—the unsub link is separate from your email's overall engagement. That said, whether or not the unsub link is tracked can depend on your ESP. Most of them don't track it in the click-through rate, but you should be able to confirm that's the case with your own ESP. Hope that helps!