A 1D game can actually be quite interesting - maybe more than we might think.
I guess the answer to our question would depend upon what we really mean by "1D". A truly one dimensional game for us has to make a choice from among three spatial and one temporal dimensions. The game that you describe uses time as the single dimension, if I understood right. A game involving only a single spatial dimension would have to be static. It couldn't progress in time. Therefore, you might not consider it even to be playable. After all, what is there to "do" and what could ever "happen", if nothing is allowed to change in time? An alternative interpretation is one that allows time, plus one spatial dimension. I think that has more promise as a "game".
While it certainly is true that there is only one spatial degree of freedom in a t+1D game, (movement is contrained to be along a "line") there is no requirement that the space in which that line exists must be "flat". In other words, the "line" can exist inside of a space of as many dimensions as you like to think of and can be all curvy and loopy. in fact, you could even have "teleportation," in which a character can instantaneously jump through discontinuities along the line, as long as doing so offers no freedom in some other direction.
In the simplest case, the path is just a straight line viewed on the 2D screen; but it doesn't have to be that. The only real rule is that the gameplay, or the movement of your "hero" character, must be constrained to the path. Another way of saying that is to say that your game must be "on rails", kind of like a roller coaster. A roller coaster can be thought of as a one dimensional experience played out in a fully 4D "reality."
Finally, if you want to go totally abstract, you can make a t+1D game in which the degree if freedom is just a parameter like invisibility (as you mention) or size or color or speed or some other parameter.
Hopefully, I have not caused anyone's brain to fall out. If so, I apologize and hope you have designated a friend or family member to accept my apology for my actions in your... absence.