Isn't it more likely that they just sent the emails themselves but used his email as the "Reply To" address? A lot of software systems (SalesForce, Hubspot, etc.) do that too.
Why? Precisely because of this: there are lots of perfectly legitimate situations when a third party sends email on your behalf.
Moreover, if LinkedIn signs their outgoing emails with DKIM, that would be a positive signal for a spam filter (and e.g. Gmail would show such mail as "sent via LinkedIn" or something to that effect).
Although "there are lots of perfectly legitimate situations when a third party sends email on your behalf" strikes me as being rather wrong. I cannot think of a single reason why anyone else should be sending email that claims to be coming from my email address. Sending email that lists me as a reply-to, sure. But as the sender? Not a chance.
Like I've uploaded version 1 of the plans, added some notes and the system needs to send out an email to everyone, I did the action, it's coming from me, not the system.
There's a reason it's part of the spec.
However, all the clients says "why does this email come from admin@thibgy.com, I want it to come from my email address, I'm sending it".
The original commented suggested that they sent the email themselves as if it had come from the user, not merely setting Reply-To.