There is a place for approximate fit in structural equation modelling.

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Sensitivity of SEM Fit Indexes With Respect to Violations of Uncorrelated Errors (2012, background): “” Moreover, misspecification search and model modification cannot be regarded as a panacea for the long and heatedly debated issue of assessing structural equation models (cf. Barrett, 2007; Bentler, 2007; Goffin, 2007; Hayduk et al., 2007; McIntosh, 2007; Miles & Shevlin, 2007; Millsap, 2007; Mulai”
Longitudinal Analysis of the Role of Perceived Self-Efficacy for Self-Regulated Learning in Academic Continuance and Achievement. (2008, background): “The posited conceptual model specifies the relations among the variables by fixing and constraining certain parameters in the model (Mulaik, 2007).”
PREDICTING MATHEMATICS ACHIEVEMENT: THE INFLUENCE OF PRIOR ACHIEVEMENT AND ATTITUDES (2011, background): “Additional evidence that the model is fitting the data is the fact that the selected GFIs generally exceed the .95 cut-off (see, e.g., Mulaik, 2007).”
A general approach to causal mediation analysis. (2010, unknown): “7 This use of residuals differs from the structural equation modeling tradition in that residuals are not being used to access the fit of statistical models as discussed in Mulaik (2007).”