Safari in macOS 10.12.2 is blinking

On iOS, HTML5's local storage is not available in private browsing mode, which means you can detect it by attempting to save something to local storage and catching the exception with the following JavaScript:

try { localStorage.test = 2; } catch (e) {
  alert('You are in Private Browsing mode');
}

^{Modified from https://stackoverflow.com/a/17741714/}

On OS X, there's not really a canonical method, but this should provide some useful information:

• Can web sites detect whether you are using private browsing mode? –Security.SE

If you're looking for implementation guidance, you should probably try Stack Overflow instead

One of the recent version update changelogs for Tweetbot states they made a change to the share sheet, which may in turn have caused a bug that is preventing Safari from showing up. This would best be reported to the developer of the app.

In iOS 8.x, a developer basically advertises "I have this type of data to be shared, what apps can share it?" The system responds with a list of apps that are capable of receiving that type of data. (Types can include text, a web URL, images, and so forth.) Depending on what type of data Tweetbot is advertising, it may not be compatible with what Safari says it can receive - Tweetbot might be advertising the URL in plain text format, whereas Safari might only be able to receive the URL specifically formatted as a URL.