If you are looking for ease of use and stability, there is an alternative solution. Issue self-signed certificates on your server, and then use a service like CloudFlare that handles the front-user certificates. The whole traffic is encrypted now, but you don't need a CA authority to deal with.
The people who are benefiting from Let's Encrypt are the least technical who can't even SSH into a server. They get the certificate from their hosting provider automagically. It's reasonable that LE and the people writing software for it will focus on them rather than another developer doing a manual integration.