Is the idea that there are just so many reasons that a value at time t is related to values at t−1 that we can't possibly control for them all with explicit parameters in a linear regression? Or is there something else going on?

Yes. If you regress the response on time (and possibly other variables) in OLS, the assumption is that residuals are independent. This might be the case for some data, and/or if you have included all relevant predictors. If residuals are correlated over time, you need an autocorrelation structure.

stats.stackexchange.com/a/35524/919 indeed provides a more general and precise explanation.

Is the idea that there are just so many reasons that a value at time t is related to values at t−1 that we can't possibly control for them all with explicit parameters in a linear regression? Or is there something else going on?

Yes. If you regress the response on time (and possibly other variables) in OLS, the assumption is that residuals are independent. This might be the case for some data, and/or if you have included all relevant predictors, and/or the effects are linear. If the residuals are correlated over time, you need an autocorrelation structure.

stats.stackexchange.com/a/35524/919 indeed provides a more general and precise explanation.