One of the most important things when you start a company is to be able to communicate with your customers and get feedback on your product. That's why the first thing I would use is https://crisp.chat/ on my landing page
I used to swear by chat widgets (Intercom). But after going full-time, I wanted to save money so it was the first thing I nixed. Over the years, I've found that email is much easier to use and is a high enough barrier to entry to stop most stupid questions and chat spam. And I got so many stupid and spammy questions that it really drove me to the point of never wanting to even check chat.
Also, I now wake up to a lot less people telling me to "fuck off." :)
I used to have one on my contact page. In total I got 2 chats in the time I received 50+ emails. Totally not worth it, especially as for one chat I couldnt even reply before the user abandoned the page.
I use them for support for various SaaS I manage at work. Really convenient for that. Use one or another maybe once every 2 or 3 months. I don't think I've ever used one on a site where I wasn't logged in, though.
I could maybe see using one on a pricing page or something like that, if I couldn't figure out which plan was what I needed.
We have one in our app, I've been amazed how users have reported so many bugs, feature requests, and tiny issues through it. Chat widget was totally worth it.
Chat widgets instantly erode my trust of your site to zero, and if you're selling anything on that site there's a 90% chance you just lost my sale. It's gaudy, intrusive, and so often a contrivance from a user standpoint that I would never add something like that to my site. A simple contact page will suffice (and require less Javascript).
I agree if it is a slick ecommerce site. But I recently went to siteoly.com to check out the Google Sheet to website capability. The chat popped up and I was talking to the creator. In a minute we were on a video call and two days later he had taken my Sheet of 2,650 vendors and turned it into a website.