What are the benefits of time-series over a well-setup linear regression for forecasting? EDIT: I just want to note that both David and Richard's answers (and comments) I think are needed to paint the full picture in response. To me I still haven't felt a real need to try out time-series frameworks from the given replies if all I can about is strong predictions and have a good sense of how to set up the regression formula, but I do take the points taken on interpretation of the coefficients and overall efficiency (and that, perhaps, I'm doing it the "hard" way).

I'm a bit naive to time-series related models like ARIMA as I can't seem to find a justification for them compared to a well-setup regression model for forecasting. Numerous responses online point to the vulnerability of linear regression due to thinks like autocorrelated errors, seasonality, and extrapolation, but it seems to me I can accommodate much of this with good data prep:

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*Seasonality - Model it. If there are periodic dips, I've had good luck catching them with various flag columns (month, quarter, even years, etc).

*Autocorrelation - Include lagged values. Taking deltas, rolling means, and various statistics against prior values seems to help nicely.

*Extrapolation - I don't see how time-series approaches navigate this any better since rare to no series is truly stationary into the future. I've found that keeping a running count from an initial reference point (i.e. months since start) seems to help general directional trends, and at some point you need to retrain anyways.

Adding on to that, regression models allow for a standard means of including  multiple other input features (for instance market data if predicting sales) that do help that extrapolation quandary, as well as the benefits of a typical "optimize it to heck" framework of machine learning... I've never found the justification for a true time-series route.
Can someone help explain what I'm missing in practical use case terms? Or is it perhaps just that regression models do require that careful data modelling before use?
 A: It is somewhat of a false dichotomy that one has choose between using a time series framework OR a regression framework.
The main reason for wanting to use a time-series framework would be autocorrelated errors. With this in mind, one can incorporate autocorrelated errors into regression by using Regression with ARMA errors (The ARIMAX model muddle - Hyndman) and in a sense get the 'best of both worlds'.
A: Autocorrelation
Here is a counterexample. Suppose the data generating process is either exactly an invertible MA($1$) model or well approximated by one. If you wanted to approximate it by a time series regression about as well, you would need an AR($\infty$) model. This is not feasible, so you would end up with an AR($p$) with some large $p$. Estimating the $p+2$ parameters* of the AR($p$) model will make it have high variance and thus perform poorly in forecasting. Meanwhile, you could use an MA($1$) model instead. It has only 1+2 parameters* and thus much lower variance and better forecast accuracy.
Seasonality
Here is another counterexample. Consider a parsimonious SARIMA model as a data generating process or its close approximation. The seasonality generated by a SARIMA model cannot be modelled by dummy variables (flags as you called them). Moreover, approximating it with a time series regression might take a lot of variables and bring in a lot of unnecessary estimation variance.
Even in the very simplest instance of SARIMA(1,0,0)(1,0,0), the SARIMA model will have 2+2 parameters* to be estimated while an equivalent autoregression (which you can consider as a time series regression) will have 3+2 parameters. This way you would be estimating one superfluous parameter and thereby increasing the variance of the model. If we were to add some moving average terms, this would get significantly worse.
Extrapolation
If by that you mean forecasting into the future (beyond the available data), then the above two points apply. Otherwise I agree that there are challenges to both approaches.
Regarding including additional variables, regression with ARMA errors (as already suggested by David Veitch) is always an option. But these additional variables typically also need to be forecast, which brings us to vector autoregressions (VAR) and VARMA models.
In conclusion, it is good to know both time series regression and ARIMA-type time series models. There will be situations where the former is more natural or more effective, and there will be situations where the opposite is the case. You can then only benefit from having both tools under your belt and being able to use whichever one works better at the given task.
*+2 comes from the intercept and the error variance.
