Am I okay in not using EC model when series are co-integrated?

I’m working with some panel data, and I’m interested in estimating the parameters of the following process:

$$Delta y_{t+1} = alpha + delta t+beta_1 ln y_t + beta_2 ln x_t+epsilon_t$$

Where $y_t sim I(1)$ (and $Delta y_t sim I(0)$). Hence, we have a cointegration relationship:

$$alpha + delta t+beta_1 ln y_t + beta_2 ln x_t sim I(0)$$

Similar work approaches this estimation problem by using error correction models using the above relationship as the co-integration relationship. Why should I do this, as opposed to simply estimating the model directly?

My intuition says that the direct approach should be valid because both sides are $I(0)$ to begin with.