By my personal interpretation, if a skill allows you to do something you otherwise wouldn't be able to accomplish (automatically or with a die roll), it's mechanically relevant. However, Arcana and Religion aren't hard wired into any core RAW systems beyond ability checks, so if that doesn't meet your standard, I suppose they aren't mechanically relevant to you.
Maybe we need a better definition of when to use knowledge skills? The PHB guides to call for checks when something has a chance of failure. That's circular because - when we make something a check - it has a chance of failure. The DMG offers a more robust guide - make a check when failure has a meaningful consequence. Knowing or not knowing the spell when thinking about Counterspelling has a meaningful consequence. So for me it's a textbook case.
At best, I would say the rules typically state when the player is not privy to certain information regarding a class ability modifying or interacting with another feature. Otherwise, it generally assumes the player knows the relevant information. Not RAW, but consistent with established design within 5e, IMO, YMMV, etc.
If nothing else, Crawford's house rule allows a caster to automatically identify any spell (at its base level) being cast if it is on their respective spell list(s):
https://www.sageadvice.eu/2016/03/0...the-spells-level-of-the-opponent-spellcaster/
At this time, however, RAW is nebulous so my presumptuousness was unwarranted. My apologies. I blame it on today being a rainy Monday and a notable lack of sleep.
Hey don't worry about it. I hope your Tuesday is much better! Let's look at Crawford's rulings...
1) If the spell is on your spell list, you identify it. So a Wizard can't identify Hex. I don't hate it, but I can't tell you precisely which spells are on what list. Not in the midst of running an encounter. I'd rather throw dice than open the book. Also, do I have zero chance of recognising a spell that I've seen the party Warlock cast a million times? Or... do I get an Arcana check?! If I get the check, why the heck not just use it ubiquitously? I think we're forced to include zero chance is right: no check allowed.
2) You don't know the level it is cast at. Again, I don't hate it. You know a Fireball is coming, but what level slot do you want to use to try and stop it. A lot of Counterspells are going to fizzle this way. Is that good? Again we have to ask, is there zero chance I can know the slot level? If there is a chance and we call for a check, why not just use it ubiquitously? Because players will foreseeably always want to slot level. Again, we're forced to include zero chance is right: no check allowed.
3) Crawford is silent here on metamagic.
Fundamentally, my goal for Counterspelling is to produce information hiding and asymmetry. Crawford's rulings do that, so to that extent I'm fine with them. Do they do that better and faster than throwing dice? Depends how good your memory is. Should casting level always be hidden? Not sure, but it could work great that way: knowing the spell and not the level could create tense decisions.
Passive to know the spell name, active to get the slot level (and name if you don't already have it). Saves a bit of looking up spell lists. Might mean you sometimes have the name but not the casting level. The two methods are pretty similar. Rolling delivers on a secondary objective (knowledge skills are useful). It's a close call.