KindaSortaNT
Sorry that you're struggling with this. I don't know that I'll have much useful help or advice for you, but the process for me was as follows.
Years ago, I read about the 'signs' of autism. I was young. I noted that "I sound so similar to these descriptions of people with autism", but determined that I didn't have it. I decided I was basically like 'an autistic person without the autism' which, now that I know more and understand that autism is a label and not a separate thing, was frankly a ridiculous viewpoint. But, when you see media portrayals you're inclined to see autism as being a very specific, exaggerated 'character'.
I'm married. I have a child. My self-diagnosis process actually started when my daughter developed fears of so many sensory things. She started having meltdowns. I remembered that meltdowns were common in autistic children, as were sensory difficulties. So, I started looking into the associated traits in more detail for her...and discovered that they almost all described me and that autism wasn't some separate thing. For reference, my daughter now doesn't show these signs much at all, though her personality is VERY like mine nowadays and so I wouldn't be surprised if she's picked something up from me. But of course, she was the trigger and that started me on the path to self-discovery.
The process has involved a lot of research from authoritative publications, it's involved the diagnostic 'tests' (notably the ones included in the AAA (Adult Asperger Assessment) and a few others), and most importantly it involved joining communities - including this one, which is where I've been the most settled. It was the last one I joined and it's now the only one I regularly partake in. By joining communities I was immersing myself, and discovering more about the little intricacies and how autistic people described their experiences first-hand, which is incredibly valuable for helping you to recognise aspects of yourself that you don't quite appreciate from a 'list of symptoms'. For example, whilst I knew about meltdowns, I didn't believe I had them often, if at all. Someone on here mentioned shutdowns, and I realised that I have those regularly and they're often described as another version of meltdown. You learn about things that aren't in the 'stereotype', but that are experienced by real autistic people. And understanding the reality is the key, I think.
As for how I'd feel if someone else suggested it? I really don't know. I want to say "I'd be grateful". Retrospectively, I wish someone had suggested it years ago and I could have been dealing with it instead of not knowing. Retrospectively, I think it's important for people to have the self-understanding and if that involves doing some research and realising that they fit the 'criteria' (or that they don't, and can move on). However, I think if I'm being honest it would have hurt to have my husband say "Do you think you're autistic? I've been looking into it". Partly because, prior to this journey of self-discovery, I was so against labels and (I confess) had something of a negative view of 'neurological conditions'.
I think I would suggest it, if I were in your position. But, yes, if you feel comfortable speaking to his family, start there WITH him once you've discussed it. They might give you/him insight into what he was like as a child, and if there's anything that stood out. I think the best thing you can do is do your own research in depth, be sure that you understand what you're suggesting before you actually do suggest it, and then perhaps mention communities like this to help him to find out more.
He knows himself well. If he immerses himself, I'm sure he'd soon work out if he didn't think the descriptions and experiences fit him, just as soon as he'd work out if they did.