While studying Panel data analysis I came across the following reasoning. We have the following general model $$ Y_{it} = \beta_0 + \beta_1 X_{it} + v_t + \alpha_i + u_{it}$$ where $v_t$ is the fixed in cross-sectional variable and $\alpha_i$ is the fixed in time variable. It is said that since $i = 1, \dots, N$ and $t=1, \dots, T$ and T is much smaller and $N$ much larger, we could introduce dummy variables for $v_t$ term, but doing the same for $\alpha_i$ would be putting too many terms. The way to address it to introduce models like First difference and fixed effect estimators.
My question is, if interpretability and computation is not an issue, what is the problem with using dummy variable approach? That is, if we still use dummy variable approach would we have biased, inconsistent coefficients and predictions under any circumstances which FD and FE estimators would avoid?