It sounds like you are part of a pretty small business given that you're an engineer who knows his customers by name. It may very well be completely acceptable to your customers to have extended downtime, or a button that disappears during a deployment leading to a broken feature for a couple of days until you, or they, realize something is wrong.

Many other companies do not work this way. If you're a huge company like Apple that makes $40M per hour, that type of downtime isn't acceptable. If you're in finance, that downtime doesn't work. If you're in healthcare, defense, aerospace, etc... that downtime doesn't cut it. Yes, you'll still have downtime in those industries from a bad deploy, but you put more checks and balances in place to reduce the odds it will happen.