Why are links missing when using the Notify module?
Your question is pretty incomplete, so let me start with adding some crucial information that I think is missing in it:


Relative path admin/config/system/mailsystem is not something that gets added to a site by the Notify module. Instead that path is what gets added after you enable the Mail System module, at least for the 7.x-2.34 version of it, which is what the rest of my answer assumes you're using.
After you enable the Mail System module you can navigate to the admin/config/system/mailsystem path.

After you do so, within the settings for "Site-wide default MailSystemInterface class", you'll be able to select either DefaultMailSystem or TestingMailSystem. However you can not, repeat not, select HtmlMailSystem (as in your question).

In order to be able to select HtmlMailSystem, you first have to enable the HTML Mail module. Which is what I did, I enabled the 7.x-2.65 version of it, which is what the rest of my answer assumes you're using.


With the above crucial additions, I think that the actual answer to your question can be found within relative path admin/config/system/htmlmail, where you can configure the settings for the HTML Mail module. I'm not sure if it's the only solution to answer your question, but pretty sure that you should select some other text filter within Post-filtering (located near the bottom within Step 3): by default it uses unfiltered, so check the results you get if you use any of the other filters you have available (or define a new filter to make it work as you want).