On the website for a Chrome extension I sell, 90% of my support requests (mostly from paid users) come in through the chat widget. The low latency communication compared to email means I can solve support requests faster, with less effort on both sides and the sender ends up happier at the end.
Having to receive, read, write, wait etc. over several emails is really painful and sucks up time on both sides even for simple problems and especially for complex problems. It can take several emails to understand a simple problem because most users aren't that precise e.g. "I upgraded but I can't access the paid features". I'm considering pushing all users to use the chat widget by default for support queries because of this.
I wouldn't be against using some chat bot automation either (which people also complain about) to deal with common requests for the initial request message because it's really hard to 100% eliminate FAQs being asked. Especially if you're a solo founder, you want time savers like this.
This is exactly why I created a bot  that blends chatbots and Stack Overflow together. You can have an automated first-line support system without scripting or conversational dances.
- Instead of conversations, you can reuse previous question/answers.
- Instead of guessing FAQs, you can let your users define them for you.
- When questions don't exist, they get catched in the traditional support flow (email). Then overtime, this effort declines.
Most chat widgets are set up to forward chat messages via email if there's nobody around and both sides can jump back on the chat widget later if they're both available at the same time which is still an improvement over just email in my opinion.