Using a sling doesn't take two hands. Loading the sling takes two hands.But while you've placed all these doubts on the evidence I've worked to gather, you still haven't shown me anything in the RAW that suggests slinging requires two hands. They had a great opportunity in the weapon table and another one in the ammunition description, and they left out "two hands" in both, obviously quite deliberately in the first case. As far as I can see, there is no ambiguity, it's something you've entirely read in on your own. It's your house rule, and you're welcome to it.
As far as rules support...
The Rules have purposely called this a Sling because we all know what a sling is, how it operates, what it looks like, etc. The rules are designed upon the premise that we have knowledge of the items they are using. You cant use a sword and say it looks like a stick, nor a horse and say looks and moves like a snake.
When the rules use the term 'tree' it is because we know what trees are, how they grow, what they look like. The rules don't have to tell us that a forest is full of trees.... that is already defined and covered by using the terms 'forest' and 'trees' in the Rules.
When the rules use the term 'sling', it is because we know what slings are, how they are made, how they operate. The rules don't have to explicitly tell us that you need two hands to load a sling....that is already defined and covered by using the terms 'sling' and 'loading'.
Now, to head off the obvious retort....some terms are redefined within the rules, and of course that takes precedence. But that is not the case here.
Oh, and it helps that Crawford has tweeted several times the same thing....