Are Hamilton’s equations time-independent from it’s general derivation?

The general derivation of Hamilton’s equations involve the change in the Hamiltonian and consequently the change in the Lagrangian that is a function of $q$ and $dot q$, $L(q,dot q)$. This is shown in this simple video
https://www.youtube.com/watch?v=jXu6zIItnLM

When deriving Hamilton’s equations for $dot q$ and $dot p$, the Lagrangian is only a function of $q$ and $dot q$ but not time. Therefore the Lagrangian used to derive Hamilton’s equations is time-independent as it appears. Does that means that Hamilton’s equations for $dot q$ and $dot p$ are only right for systems for which the Lagrangian is time-independent (and therefore the Hamiltonian – or energy is conserved)?