Comparing results from structural equation model and OLS regressions Are results obtained from a structural equation model (SEM) comparable to those from a series of separate regressions?
Or, conversely, can I obtain direct, indirect, and total effect of an independent variable of interest by running three separate regressions (e.g. OLSs)?
Let us say that IV=independent variable, MV = mediation variable, and DV = dependent variable. I conduct three separate regressions.

*

*DV = B0 + B1*MV + B2*IV + Controls
B1 is the direct effect of MV on DV, while B2 is the direct effect of IV on DV.


*MV = K0 + K1*IV + Controls
K1 is the direct effect of IV on MV.


*DV = P0 + P1*IV + Controls
P1 is the indirect effect of IV on DV.
The notation used in the mediation analysis below is based on that used in this handbook:
Hayes, A. (2013). Introduction to mediation, moderation, and conditional process analysis: A regression-based approach. Guilford Press, New York, US.

*

*K1 = a is the direct effect of IV on MV

*B1 = b is the direct effect of MV on DV

*B2 = c' is the direct effect of IV on DV

*B2+B1*K1 = c'+a*b is the total effect of IV on DV

In turn, the total effect, that it, c'+a*b, should equal the coefficient E1 from the following regression:
DV = E0 + E1*IV + Controls
To my understanding, a*b is a measure of omitted variable bias. Thus, the entire SEM could be seen as an application of the Frisch-Waugh-Lovell theorem (Greene, 2003; p. 148-149).
However, when I compare the results obtained from this series of OLSs and the results from an ad-hoc package (i.e. I use the built-in GSEM in Stata), that results do not compare. In particular, while "a" and "b" are similar, "c'" is very different.
Here is a related post on Statalist, where I discuss the script. While the question is more focused on the script per se in Statalist, here I focus more on the methodology.
[I am new to SEM]
 A: The answers to my questions are "yes".
This was the issue on which I have wasted a couple of days. The independent variable is endogenous, so I have regressed the first stage and obtained the predicted values of this variable (I had omitted this detail from the post).
Then, I have used the (predicted) IV so obtained in both methods, that is, series of regressions versus GSEM; however, the first stage to obtain the IV used a slightly larger sample. The reason is that the mediator variable (MV) has some missing values.
My solution was first to run a rubbish OLS: Y IV MV Controls, and flag the sample.
Then, I have regressed the first stage to obtain the predicted values of IV.
Finally, I have used these predicted values in both the regressions and in GSEM. Now the results are identical across the two methods.
The lesson learned is: make sure that the samples used across methods and different stages are the same.
A: 
Are results obtained from a structural equation model (SEM) comparable to those from a series of separate regressions?

Yes, unless the SEM places constraints that are not in place among the separate regressions. Such constraints (especially when invalid) can systematically bias the estimated effects of interest.  In SEM software, the optimizer tries to find a set of parameters that reproduces the whole covariance matrix (and mean vector), so "propogation of errors" is a problem.  Some estimators are more robust to this (e.g., SAM or MIIV-SEM)
https://osf.io/pekbm/ (SAM preprint)
Bollen, K. A. (2019). Model implied instrumental variables (MIIVs): An alternative orientation to structural equation modeling. Multivariate Behavioral Research, 54(1), 31-46. https://doi.org/10.1080/00273171.2018.1483224

Or, conversely, can I obtain direct, indirect, and total effect of an independent variable of interest by running three separate regressions (e.g. OLSs)?

For OLS, yes (again, assuming the SEM is unconstrained: df = 0).  This does not generally hold for generalized linear models:
Breen, R., Karlson, K. B., & Holm, A. (2013). Total, direct, and indirect effects in logit and probit models. Sociological Methods & Research, 42(2), 164-191. https://doi.org/10.1177/0049124113494572
