Note that all $2021$-tuples $(a_1, a_2, \dots, a_{2021})$ of positive integers whose smallest element is at most $3$ such that consecutive elements differ by at most $1$ can be uniquely defined by $(b_1, b_2, \dots, b_{2020})$, where $b_i = a_{i+1} - a_i$ for $i = 1, 2, \dots, 2020$ and each $b_i$ can be $-1, 0$, or $1$, and the smallest element of the tuple, which can be $1$, $2$, or $3$. There are a total of $3 \cdot 3^{2020} = 3^{2021}$ such tuples. But these include tuples that only include $1$s and $2$s, of which there are $2^{2021}$ of them. Thus, the number of tuples that satisfy the problem condition is $\boxed{3^{2021} - 2^{2021}}$.