The Hubble constant is not necessarily derived from the cosmological constant. To be clear, it's not even a constant either, it's a proportionality factor between the recessional velocity of distant objects and their distance from us. Though in some usages it refers to the value of this factor at the current time, which would in fact make it a constant. Regardless, even without a cosmological constant, after the Big Bang you'd still have an expanding universe, possibly collapsing or endlessly expanding, and I like to think of the Hubble constant/parameter in this case as representing the "momentum" the matter in the universe has left from the Big Bang.

The cosmological constant does in fact have to be constant within the constraints of general relativity. The mathematical machinery of GR only allows two parameters: Newton's constant G, representing the coupling of matter to gravity, and the cosmological constant. Both have to be true constants, numbers with a unit.

However, this is only true of the most basic theory of dark energy, where you directly add a constant to the general relativistic lagrangian. More complicated theories, like, for example, quintessence, involve adding new dynamical fields to the theory. The "effective cosmological constant" associated to such theories, quantifying the effect these fields are having on the expansion of the universe, can dynamically change over time, and some of these theories are proposed to solve the Hubble tension. To be clear, although none of these theories are fringe or pseudoscience, they haven't been accepted as a final explanation either, resolving these issues is still a work in progress, at this point they're all simply interesting hypotheses.