Should I use coordinates as features?
There is a couple of separate questions I can identify here: one is whether it is appropriate to include coordinate features and the other is whether it is appropriate to convert a multivariate regression (more than one outcome variable - 9 in your example) into a regression with single dependent variable.

Coordinate features

If you think that the dependent variable value in a particular cell of your grid is somehow affected by the location of the cell within the grid (in other words, that values in the same cell are correlated), then you should probably include coordinate features (which can be simple cell identifiers or something more complex, like a more specific identification of where the cell is located within the grid relative to a particular point).  It is possible to even use target-based encoding (i.e., use the mean of dependent variable values in the same cell as a new feature), though that gets into issues like data leakage and overfitting (which can be addressed separately).  You may also possibly want to create interaction terms, allowing for the effects of non-coordinate features to vary by level of a coordinate feature.

This is different than modeling correlation between values across different cells.  If you expect that there is some spatial autocorrelation between cells in this 2-d grid, e.g., a value in cell (1,1) is correlated with values in cells near it: (1,2), (2,1), (2,2), you should probably model that relationship by including some set of spatial features (not just coordinate identifiers), like spatially clustered covariates, or by using a spatially lagged dependent variable model.

If you are not sure if spatial correlation is present, you can estimate a model without any spatial features and then test for spatial autocorrelation (spatial dependence between residuals of regression) using Moran's I (a positive Moran’s I hints at data is clustered, a negative Moran’s I implies data is dispersed.)

Multiple vs single dependent variable

If you go with option 2 as you described it, with one output variable, you will be modeling a completely different variable than if you were modeling 9 (in your example) separate variables.  Each of the original 9 variables likely has its own statistical properties, but if you combine them into one, you will have a new variable that has its own properties (some combination of the 9).  So if you consider the grid to be 9 separate variables, you should stick with modeling 9 separate variables.