Thanks for following up and giving the button switch a try. I've read thru some of the other suggestions here and would have thought that one of those should have solved the problem.
I did some add'l testing this morning and setup a Guided LP template with some variables to match your situation. What I discovered is that when the variables are contained inside an editable text area that they don't work, but when they are not inside an editable area, they do work.
Here's a few images for reference of the testing setup:

(above) There are two buttons here (same button) -- one is included inside an editable area, the other is outside the editable area

(above) The highlight parts are inside an editable area (and do not work) and the bottom button is outside the editable area (and does work).
Is there a chance that you're using the variables inside an editable area on the template?
If so, I think that could cause the issue you're running into here. If that's the case, you really don't need the variable there b/c you can use the Rich Text Editor to go in and change the URL and Text of the button. If you use the <button> element, it'll be harder to edit the buttons using the RTE but if you use the <a> element, you can use the Insert/Edit Link tool on the RTE toolbar to easily edit the button text and link. I'd recommend using the <a> tag over the <button> tag in this situation.