If you're reading this and need a short escape from life, I really recommend looking into doing the trek to EBC. It's extremely accessible (and cheap), and I've personally seen people as young as 8 y/o and as old as 80 y/o doing it and having the time of their lives.
I've been all over the world. Antarctica, climbing in Pakistan, trekking through the Middle East... Nepal was one of my first adventures and remains one of my favourites.
To stay on topic... Teahouses can be found along every route through the mountains. Most of my fond memories are in teahouses, where you get to sit, have a black tea, and reflect on just how beautiful the people (and the mountains) are. It's also a great opportunity to meet people from other cultures. I'll always remember the Russian's who bought me a beer after climbing Island Peak, the cards I played with some kids in Samagaun... I could go on.
Go visit Nepal!
From there, you either hike up the Kala Patthar view-point which has that famous panoramic view of Everest, basecamp, and all the surrounding peaks... Or you trek around the canyon and begin the approach to EBC itself.
There is a very nice variant of EBC hike called 3 passes trek. Goes over 3 high altitude passes (5500, 5400, 5300m) on top of base camp, making a nice loop. A better challenge, once outside main valley just few people, and views are stunning, ie from Gokyo Ri, or north walls of Taboche or Cholatse. A difference between meh and stunning for me.
It was an incredible experience, but not for the faint-hearted. A couple of people in our group were unlucky and had serious issues (oedema and an aneurism) and were too high for helicopter evacuation, but they both survived. If you want to do something like this, go with a reputable company (such as KE Experience in the UK).
If you are significantly overweight or have dodgy knees, then it isn't for you. You are moderately fit and prepared to do some long walks in preparation, then you are probably fine. If you aren't sure, maybe go on one of the lower level treks and see how you get on.
Aconcagua in Argentina. It's on my 'things to do if in the area' list. The nearest airport is only a 200km drive on a tarmac road away and then a 40km trek to the peak! Although the 4km altitude gain is likely to be harder than the distance.
BTW Everest is so steep that it doesn't have a lot of snow on it compared to other high mountains. So it is not even a very attractive mountain (subjective obviously).
That altitude was about my limit even at the time. I had done a couple of previous Nepal treks to fairly high altitude but not actual climbs and had done a lower altitude glacier climbing school in the US northwest.
You just pack clothes, no matter how remote your destination is, there’s going to be food and shelter available every 6-8hours.
In Nepal? That sounds like a risky assumption to make.
Of course if you go completely off-trail for days all bets are off.
Maybe highway rest-stops are the closest analog for the US but even many of those have been shuttered by governments driven to parsimony.
The word for it is ‘home-stay’, there are a few houses in every village that are set up to accommodate guests for a very reasonable amount of money.
And these villages are pretty much everywhere.
I have been lost in the Himalayas, and it was not that much work to walk down the river to a village.
I did find, like it seems you did, that I loved traveling through Nepal and the accommodations you've described. Remarkable and tough people living hard lives with resilient cheer.
Nepal and Bhutan has less population, so the density of humans vs nature is less there.
But if you stick close to the hiking trails in this vast & beautiful subcontinent, you can backpack easily from one rest stop to another.
And after some hours of tough trekking, when you encounter the warm welcome, bonhomie, food & drinks, music & laughter from the natives/locals, it is an amazing experience, worthy to be cherished. Faith in humanity - restored!
Not being able to solo hike really ruins the experience for me, so I'll be avoiding Nepal until that changes.
In the areas that few people go to, it isn't actually enforced.
The Nepalese government just introduces this rule every few years in response to a missing trekker in the guise of safety.