I'm biased--I created the all-in-one MythTV installer for OS X.
I'm doing much of what you are interested in. I have a 2013 Mac Mini connected to my 50 inch TV. I record OTA via 2 HDHomerun tuner boxes (total of 4 tuners). No need for a NAS, I just have a couple of external USB drives attached to the Mini for storage. In addition to TV, I've ripped a few DVDs. MythTV has a built-in video manager with metadata, etc. Rock solid setup. The software never crashes. Every once in a while, bad weather messes up reception so that a recording fails, but that's it.
However, we don't watch on iPhones or iPads. You have to trade off picture quality for file size and compatibility. You have a big 4K TV so the uncompressed OTA format (MPEG2, up to 19 MBps) will look really, really good. I'm also a big fan of multi-channel audio and I've got a 5.1 setup that I really enjoy.
MPEG2 is basically not supported on phones and tablets. And wireless networks don't consistently deliver anywhere near throughput that they brag about. So the video needs to be re-encoded to X264 and re-compressed so wifi can keep up. Everybody has a different tolerance for video compression. An option for you is the HDHomerun Extend. It has built-in hardware to re-encode and compress with a few different quality/size options.
Myth also has an experimental feature to re-encode/compress on the fly and play the resulting video in a browser window on your phone or tablet. It is OK if you want to play from beginning to end but there is no commercial skip or other features.
An issue you didn't mention is remote control. MythTV supports the (gum stick) Apple Remote. Even though it only has 6 buttons, you can access every feature in Myth via menus. Newer Mac Minis no longer have a built-in IR receiver so you need to think about how you're going to control the system. You can use a Bluetooth keyboard/trackpad that doesn't always pass the spousal approval factor.
Re the AppleTV, someone wrote a MythTV client for it. I haven't heard a great deal about it. I can't remember if it can play MPEG2 content or not. From the AppleTV, you should be able to search the App store for "mythtv" and find out more.