Hey marliesB.
This is a really good question, and the theory you are eluding to is something I've been working on in my own game lately so I'll try to answer as best I can.
When we first start to learn post flop strategy we are shown the power of the continuation bet, and ever since that day it feels like the most natural thing to do. Because of this, it feels slightly uncomfortable taking passive lines with the pre-flop initiative, and you may feel like you're guessing quite a lot down the streets. These feelings are natural, and although the c-bet has some merit, there are loads of spots where taking a passive line is better and I'll try and explain why.
First of all let me respond by saying that the two situations you mentioned in your OP are very different and should be treated as such. In this reply I am going to deal with the former.
Let's say that we open Ah Ts at $100nl for $2.50 in the cut-off and a competent villain peels the button. We are both 100bbs~ deep, the blinds get out of the way and the flop comes Jh Th 4c. There is now $6.50 in there, and is might seem very natural to bet on such a drawy board to protect your hand and for value, but can we make a more profitable check here? Let's see.
First of all, you have to think about what kind of opponent you are dealing with.
Now our villain in the example is a competent reg, so he will be expecting us to check/fold this board texture a lot since it hits his "perceived range" quite hard since he will have quite a lot of middling suited connectors which have straight/flush draws and broadway hands which have smashed the board when he peels the button vs our cut-off open. Because we want to be able to check/fold this board quite a lot, we should have some hands that we are going to check/call right? So now we have to decide which type of hands we want to check/call and why. We should start by throwing some of the hands as described in the OP in our checking range because Tx hands aren't quite strong enough to bet vs a competent opponent since he won't make really bad calls and he can raise this board a decent chunk of the time and really put us in a coffin since we now have no idea what to do. We can't really start bet/calling AT and just heroing down without really specific reads, so let's just check this type of hand and call his bet.
Because he expects us to c/fold this board so much he will (most likely at this level, the following statement is less true at high levels as the standard of player increases) bet an "unbalanced range" when we check. What this means is that he won't bet a range of bluff hands which is proportional to how many value hands he has; he will bet almost anything he's called with pre-flop because of the reasons I stated above about his perceived range. This makes c/calling a very strong and profitable play and should be something we do with our weaker value hands where we can't feasbily get three streets of value (weak top pair hands, second pair hands against anybody even slightly aggressive).
In our example we have a perfect hand to do this with because we gain so much more value from his bluffs than we could get from weaker Tx hands if we bet since there are so many more combinations of bluffs relative to the hands that we beat. Also, we have the Ah so we can continue on every heart turn too, which really increases the playability and durability of our hand.
Now obviously there are downsides to this play, in that we ARE giving him free cards to his draws and overcards, and we also allow our opponent to value bet Jx for two streets at least, since as you very correctly stated in the OP, everyone seems to be doing this with their weaker showdown hands atm. So how do we counter-act these downsides? We add strong hands into our c/calling range.
On boards where you want to be able to check/call weak showdown hands and not get crushed, throw some strong hands like top set and AJ/KJ in there and be prepared to call down three streets and check the replayer to see him showdown some ridiculous hand that he has decided to go crazy with because he thinks you have second pair. Also, he will always bluff the cards which improve your hand when you have KJ/AJ since he never thinks that you have any other overcards other than your kicker to your perceived second pair hand, adding even more value to the check/call.
If the villain was a passive "fish" or nitty reg then if we checked they may take a free card with a lot of the weaker draws that they would call a c-bet with, so we would be losing value from our hand and opening ourselves up to getting outdrawn for free by checking, so betting would be a better option. Remember that you don't have to only apply this logic to your weaker hands, you should consider checking three tens here (or similar strength hands) but for the reasons I've stated above, end up betting for value.
I don't want to get too technical ITT but if you do want to start check/calling your weak showdown hands in cash games, then i do genuinely believe that it's VERY important to have a somewhat balanced range, even at $100nl and similar levels. So you want to have a balanced check raising range and a check/calling range which involves both strong and weak hands. I'm not going to break this down right now because this post is already a bit TL;DR and I could spend pages and pages dissecting checking ranges on different board textures. Also you can break this down mathematically to work out what % of your range you should have etc. but that is very involved and this is just a general theory response.
So let me summarise.
In spots where you are out of position post flop and have the pre flop initiative, we should have a check/call and check/raise range on boards where we are expected to check/fold a lot (ie boards which are good for our opponents perceived pre flop range). We do this because it allows our opponent to bet TOO MUCH so we can make a considerable profit. As a general rule we should be considering check/calling with the hands where you can't get three streets of value, so we gain more value from his bluffs (ie second pair and weak top pair hands). If we want to do this (which we do) then we should be check/calling with some of our strong hands too (top set, TPTK etc.) sometimes to balance it out, and also check raising sometimes too (but that's a whole other thread).
I hope this makes sense, sorry if it's too long and a bit much to digest, i just think that it's an important part of a strong NLHE strategy.
