Erase button

The only difference between this and current methods of painless and quick suicide is how "easy" it is for such an intention and understanding to turn into an actual case of non-existence.

Building the rooms everywhere and recommending their use to anyone with such an intention ("providing" them) makes suicide maximally "easy" in this sense. On a surface level, this increases freedom, and allows people to better achieve their current goals.

But what causes such grounded intentions? Does providing such rooms make such conclusions easier to come to? If someone says they are analyzing the consequences and might intend to kill themselves soon, what do we do? Currently, we force people to stay alive, tell them how important their life is, how their family would suffer, that suicide is a sin, and so on, as a society, and we do this to everyone who is part of society.

None of these classic generic arguments will make sense anymore.  As soon as you acknowledge that some people ought to push the button, that anyone might need to consider such a thing at any time, you have to explain specifically why this particular person shouldn't right now, if you want to reduce their suicidal intentions. The fact that someone considering suicide happens to think of their family as a counter reason, is because of the universal societal meme, not its status as a rational reason (which it may very well happen to be).

We can designate certain groups (i.e. the terminally ill) as special, and restrict the rooms to them, creating new memes for everyone else to use based in their health, but the old memes remain broken, and the new ones may not be as strong.

I suspect that the main impact of providing the rooms will be socially encouraging suicide, regardless of what else we try to do, even if we tell ourselves we are only providing a choice for those who want it.

if possible, should such a room be provided?

Informal argument: no. People's perceptions are noisy. On the short term they fluctuate far more than is strictly rational. I believe that most people's views of existence on average are (weakly) positive. With the button, if/when there's a momentary fluctuation...? A positive mean does not imply all data points are positive.

I would consider the existence of such rooms to be very dangerous.

Not for their stated purpose, which doesn't interest me much from an ethical point of view one way or the other. Their existence implies the prevalence of technology that can instantly read minds well enough to reliably determine both intention and understanding of potentially far-reaching consequences of carrying out that intention. While such technology could be employed very usefully, it also seems very easy to misuse.