ASK Him - do you really know this CEO or is he just another one of your 10,000 Linkedin connections?
ASK The Headhunter to foster an appropriate introduction. Give him say 10 business days to line-up something up. Be transparent about this so there's no misunderstanding.
If he can't or won't -- then, practice courage and email/call the CEO directly.
Surprisingly, they often admire people with the chutzpah to say -- here's why you & I should talk.
If you can't find their direct contact email I'd simply send them a connection request on LinkedIn with a short message "Hi, I really like thing about company and wanted to see if we might be able to chat about potential opportunities for me there".
Worst case they ignore the request.
Back in the day I recall searching for a job and having little success. I found one job with a company I thought was perfect. I submitted a resume. A month late their HR called me up and told me 'I found your resume searching online and I think you will be perfect for a job at my company'. He described the job I had applied for. I agreed and explained it was the same job I had applied for ... never heard from them again. Why? They get paid to find YOU. The HR department does not look o brilliant if you find them. I know this is a very negative opinion of HR, bit it seems common.
For some reason HR departments are mystified by IT requirements / resumes. So they pay services to filter your resume. But they trust the headhunter who bought them lunch / dinner to know what they are talking about. Also if you do not work out they can blame the headhunter.
You have nothing to lose by using a headhunter, unless they try to charge you.
Once again, especially since I just reread your story, it is an obvious YES.