I am a postdoc and will soon(er or later) begin a faculty position. At the moment, due to a rough academic job season and an increasing sense of feeling limited in what I can do in my current postdoc, I am feeling very closed off from others in the field and a strong desire to make connections. Today I had the idea of inviting myself to give a talk in a lab that does research I am very interested in. I am not seeking a position in the lab but rather just want to make connections with others, learn more about the work done in the lab and how it's done, and suss out potential for future collaboration. It is the 'off season' for conferences in my field and I don't have work that I am ready to present in a conference setting, but I do have a talk that's ready to go. Would it be weird to offer to give a talk? I would pay my own travel/hotel expense. It wouldn't be a complete cold call because we have a mutual colleague that I could mention.
There's absolutely nothing wrong with making an outreach to a group you're interested in learning more about. Just make sure not to make it too hard of a sell: you could say something like, "I'd like to visit your group, and I could give a talk if you'd like." I don't think most professors would be put off by such a request.
It is usually hard for institutes to find external speakers (and expensive to invite), so anyone willing to visit at their own expense is warmly welcomed. In particular if the institute has a (semi)-recurring seminar series to fill with speakers.