How long do I have to submerge an egg in a specific temperature water to boil it?
In theory you should be able to reach your goal, sort of, borrowing from the Japanese Onsen egg and the Sous Vide technique. Serious Eats has an article detailing the results of slow-cooking eggs at different temperatures just around your given 70 C for further reference. You should aim to keep the water around the egg as high as possible and do so for the better part of an hour.
But let’s be honest - even with constant re-filling of 70C water you won’t end up with a solid egg and every degree lower will give you more and more watery whites - which I suspect you mean when you write “gross”. So while you can safely eat the egg, it remains to be seen whether you want to. And it’s a huge hassle, just for a soft egg.

But let me suggest an alternative, using the microwave you mentioned. Eggs in a microwave are doable, if you take the egg out of the shell. If you are really limited to a glass - assuming it’s a plain and reasonably heatproof one, otherwise it shouldn’t be used in your original setup either - use the glass. Pour about half a cup of water into the glass (or small bowl, if you can find one, for a nicer shape), add a generous sprinkle of salt. Crack your egg into the water and microwave the whole thing for about a minute to the desired doneness - no need to actually boil the water, actually you want to rather avoid boiling, but it’s not the end of the world if you do get bubbles. This should give you a nice poached egg. Zapp it a bit longer for a solid yolk. Of course, each microwave is different, so a bit of experimentation is in order, but this method should get you closer to your goal than the water dispenser one you were considering.