So with the help of others above, I finally found a better solution. That require no brutal fix to re-adding title to every single elements. Don't get me wrong, that fix will work, but performance is important (especially I still need to support IE8 ugh).
Basically I add a custom variable to the tooltip object, this can also be a global variable. Since everything is an Object in js, you can just add anything you want to it.
$(document).tooltip.temporarilyOff
Then when I initialize the jQuery tooltip, I just need to add a check in the open:
var settings = {};
settings.tooltipClass = "tooltip";
settings.open = function (event, ui) {
    if ($(document).tooltip.temporarilyOff) {
        ui.tooltip.stop().remove();
    }
};
$(document).tooltip(settings);
Then when I need to temporarily disable the jQuery tooltip, I just need to toggle the flag anywhere I want. Like so:
$(document).tooltip.temporarilyOff = true;
Anything after this point, the tooltip won't be triggered, and all the elements will keep their title attributes. When I am done with what I am doing, I just need to set the flag back to false and tooltip will work exactly like before.
I can probably make this into jQuery plugin for easier calls and also hide the slightly ugly variable name... but anyway that's the idea. I think this is a much better fix as it won't force jQuery to remove the title attribute for nothing, then adding it back afterward doing twice the useless work.
Here is the updated example forked from @Jasen's original jsFiddle: