I will also say we use live chat extensively in our product and it's nothing but great. We have a sub 1 minute median response rate for all chat requests, and our customers love it. Great support matters, and chat is great support.
When the person replying on chat is knowledgable and is able to reply quickly, it's a great support experience on both ends I think.
Automatic pop-ups, bots replying instead of humans and unhelpful support staff is what gives chat a bad reputation but when used right I think chat can be an order of magnitude improvement over email.
Email can be used badly as well - I find some large companies take several days to reply to support emails and have staff that mostly reply with templates triggered by a few words in your message instead of understanding what you're asking.
I've been going back and forth with Google on an oauth consent screen approval for 2.5 months now, all over email, with multi-week gaps between each response from them.
I'd give anything to just hop on a live chat with them an answer any questions they have.
In the event that I can’t fix their issue very quickly (minutes), I usually tell them I’ll follow up here and over email to let them know I’ve fixed the issue or addressed whatever they were raising. Because users of our product tend to have us open for a long period of times (companies use us for organization wide 360s), I usually can catch a person why they are still in the product.
As for juggling all the other things you’re working on as a startup, paying customers who have a question / issue almost always trumps every other thing I’m working on at any other moment. Outside of meeting with another customer, I will push whatever I’m doing and shift my attention accordingly.
Even in that 24 hours, dealing with support via email was a distinctly crappy experience compared to a live chat widget.
I've found a chat widget really helpful for my own product (https://www.checkbot.io/) - I find the reduced lag of the chat format means problems get resolved quicker on both ends and the style of replying (e.g. maybe 16 short messages back and forth with chat via 4 longer ones with email) results in you getting more information and making more of a connection with your users.
I use the free tier of https://crisp.chat but most of them are fairly similar for basic usage. I would say avoid using one that has that annoying "Hey! You! How can I help?" automatic pop-up feature though.