Okay, I'm totally aware that MC in such VNs should be only a vessel for the player to pour their personality in. But I have another inquisitive question/suggestion. :)

So, what we know about MC:
1. He had travel a lot.
2. He has some place, that he calls home, though it is mentioned only once. It's the place where he rests between meeting the old man and going to the hotel.
3. He has no friends or family to speak about. At least, once we have an internet access in the hotel, he's not trying contact anyone from his 'past life'.
4. He never tries to travel outside the hotel, be it to visit someone worth visiting or just trying to get some mementos from his home. It is not forbidden by the rules, as far as Master Jean have a possibility to go outside and participate in a war, and, sadly, be killed there.
5. He never mentions anyone or anything from his 'past life' in his talks with Asterion or other guests/employees of the hotel.

So yet again, I understand that 'hollow' MC is a vessel to pour yourself into for deeper immersion in the game.

However I must ask the question, if MC is not me, and I just guide him, same as the narrator is not just 'a voice describing what's going on', but someone else's words, be it Athena (god have mercy on my soul, I would despise her seeing and commenting everything I do) or some other entities... well, if me and MC are different entities, shouldn't he be as grateful as Asterion, for receiving a purpose and close friends?

Let's imagine MC without the player's guidance? It's a lost soul without a place to call home, without friends, and close relatives? He is educated in one field or another, but he just travels, taking odd jobs here and there. So, what we have here is:
1. MC's personality past/character is intentionally left ambiguous, letting the player to speak with available actions and words of MC, and when actions and words are not enough, figure everything out by themselves, even if it means screaming into the cold screen. :)
2. MC is underdeveloped, and deserves more options, to tell Asterion and/or Kota and Luke about his past, maybe express the same feeling of getting some purpose and goal in life, when meets Asterion.

Okay, summarising: I understand if MC is an empty vessel for the player to put himself in for immersion, but I find it somehow lacking of being able to express some well deserved gratitude to Asterion and other characters, talk about the past (every previous Master had their past), and maybe find more meeting points between MC and other characters. If it is the original design of Minoh team, I bow before their will and solemnly swear to play the game as it is at least a hundred times until I check every possibility and every choice, except for hurting Asterion (no one can drag me to this path!), however if it is something that could be thought of and implemented, I hope it's not a bad idea. No pressure. At all. :D

edit 1: p.s. Sorry if this topic duplicates some old one, I tried to search by "MC", "background" and "past" and didn't find anything similar.

edit 2: p.p.s. I remember that there is a drinking game, where MC can share some lewd (and not so lewd) insights from his past, but that is still not enough, especially, if we don't know, will choices in the drinking game have some consequences, in... ahem... more private relationships with Asterion. :)

I feel like some of the background or daily goings-on with the MC, like calling/interacting with family or friends outside the hotel, or lack of detailed info about his past comes down to two things.

The first is like you said, the MC is a vessel for the player to imagine themselves in  and is left vague purposefully in order to better facilitate a sort of 'head canon'.

The second option is what I'd call episodic ambiguity/ the bathroom absence. Okay so what the heck is episodic ambiguity/ bathroom absence?... Well in most forms of storytelling, such as film or aduio or VNs, the authors leave out common daily details like using the restroom. They do this for a couple of reasons, one it's so common and universal to everyone that we simply don't need to witness it/ it's a bit gross to focus on and wouldn't add to the story. This is bathroom absence.

Then you have the idea of episodic ambiguity, which is the concept of events happening outside of what the viewer witnesses. Many television shows reference people, or places, or events that were never shown on screen. This creates the illusion of a larger more realistic and developed world, without having to spend extra time creating that scene.

For this VN the meta answer to your question about missing details in the MC's background comes down to the authors not having enough time and energy to create an infinite set of options, and their desire to create a blank slate for readers to project onto.

I my own in universe 'canon', I tend to write these oddities off as a form of episodic ambiguity and bathroom absence events.