Can I use SEM to purely compute latent correlations? Do fit indices make sense in this instance? I have 3 variables X, Y and Z, with each variable being assessed by 3 items. The variables are not part of the same factor. Does it make sense to use a SEM model to purely compute the latent correlations without specifying any regression paths or dependent variables? Example using the lavaan syntax in R:
model <- 'X =~ X1 + X2 + X3, Y =~ Y1 + Y2 + Y3, Z =~ Z1 + Z2 + Z3'

fit <- sem (model, df)

I have not come across any articles that use this method to purely compute and report latent correlations. Does it make sense to do so? If it does, do fit indices make sense here as well?

Reporting (estimates/tests of) the latent correlations rests on the assumptions that (a) those latent variables exist and (b) are measured by the indicators.  Poor model fit is evidence against at least assumption (b), maybe hinting that even (a) is dubious.  So it is still recommended to report how well your model fits your data:

most often the following estimation results are of primary interest:
estimates of the fit of the model, estimates of model parameters, and estimates of the (asymptotic) standard errors of parameter estimates. The presentation of these results can be organized around four main questions... (p. 472)

Boomsma, A. (2000). Reporting analyses of covariance structures. Structural Equation Modeling, 7(3), 461-483. https://doi.org/10.1207/S15328007SEM0703_6
See also:
McDonald, R. P., & Ho, M.-H. R. (2002). Principles and practice in reporting structural equation analyses. Psychological Methods, 7(1), 64–82. https://doi.org/10.1037/1082-989X.7.1.64
FYI, your lavaan model syntax will only work if you replace commas with semicolons, or place each formula on a separate line.
model <- 'X =~ X1 + X2 + X3; Y =~ Y1 + Y2 + Y3; Z =~ Z1 + Z2 + Z3'
## or
model <- ' X =~ X1 + X2 + X3
           Y =~ Y1 + Y2 + Y3
           Z =~ Z1 + Z2 + Z3
'

