Do you follow the default rules and how many encounters do you actually get in?
I tend to think that the way most tables actually play, they should probably be using some kind of variant rest period.
I honestly don't think I've ever run a game that had more than four combats in a single day.
There's the one night long rest and the week long rest in the DMG.
In Adventures in Middle Earth you can only rest in a sanctuary (so no easy long rests during wilderness travel)
Even in AD&D you didn't just have to sleep. You needed to be rested and then you needed to memorise your spells (15 minutes per spell per spell level - which could take up most of the next day as well)
In 13th Age you just get the equivalent of a long rest every 4 combats. (The GM is free to adjust it if needed but it has no relation to time passed.) However, I suspect most tables would find that too arbritrary.
Yes I follow the default resting rules but since they include GM determining if rests can be done under circumstances that seems to work fine.
As for counting encounters per day, that's not as easy for me given my style of play. Also "day" in this context does not really mean "game dsy" and certainly not "play session."
Of "encounters that matter" it's often a case that it's what 5e calls "extended envounters" where the challenges and adversaries come in rather quick succession - with enough pauses to enable but limit short rests (and often make them risky) and keep long rests off the table (without severe costs.)
For example, the PCs decided to say the heck eith it and raid a temple... actual thriving one in a town. Thry had other option but for tolrplaying reasons they followed the one PC lead.
So the approach, entrance and VIP guest levrl too an entire week's session. Multiple skirmishes and sub-bosses that setup an opportunity to break off now if they wanted. It did not hurt that much, but did eat up more than a few of their higher dprlls to get by so quickly, limiting NPC reactions.
Second week, they went for stretch goals down the the "elite" level, had their own issues with some clean-up along the way, engaged much tougher foes and took a lot of hits, a lot of scares and prevailed with quite a bit of drama and loss.
Now, at that session end, they are still en prise, do yo speak, because they are farther from their exit, still have the threat of all the outside stuff they haven't dealt with (guards and other forces now responding to the raid) and some very difficult choices on how to handle it now, this moment. Do they grab whatever is at hand and make a break for it, maybe fighting to an exit? Do they try and bunker in for short rest, better search, etc hoping rap counter-attack doesnt come too soon? Do the fo something clever to maneuver a longer cease fire for longer rests and more options?
We are heading into our third week of play with another pause to reassess and regroup with a now quite tired party and very different options afoot.
And, fun thing is, from the GM seat, I had in my "think they might try this" partial expecting had thought they would have taken the non-raid options laid out by their informant. So, this current gauntlet is of their own choosing to largest degree.
So, in what most folks would vount as encounters this would be... about 5-6 so far, more 6 I think, very much in line with the 5 room framework with so far some pauses (three I would say) but no rests and it's not over yet. Its so far been very Three-Rs compliant.
Depending on their next choices this could easily turn into another 2-4 encounters before the get a long rest, but maybe a short rest in the very near future (costly though.)