When people say that "unitary time evolution does not change the total entropy" and "entropy increases over time", they're talking about two different kinds of entropy in the two sentences. If the entire universe is in state $\rho$ and its time evolution is given by the Schrodinger equation, then its "total entropy" $S = -\mathrm{Tr }(\rho \ln \rho)$ is constant over time. But when people talk about entropy increasing over time in a closed system, they mean the internal entropy of its spatial subsystems. That is, if you take a spatially contiguous subsystem $A$ of the entire universe and form its reduced density matrix $\rho_A$, which acts on a smaller Hilbert space than $\rho$ does, then its entanglement entropy $S_A = -\mathrm{Tr }(\rho_A \ln \rho_A)$ will generically increase in time, even as the total entropy remains constant.
More roughly, the different spatial pieces of the universe get more and more entangled with each other, increasing the "internal, relational" entropy, even as the universe as a whole does not grow any more entropic. A strange fact of quantum mechanics, which does not occur in classical physics, is that the entropy of a subsystem can be much greater than the entropy of the entire system, so this process can continue indefinitely.
All this can be done entirely through Schrodinger time evolution, with no need for collapse to be invoked at all. Depending on your interpretation of quantum mechanics, collapse may cause an increase in entropy, it may result from an increase in (entanglement) entropy, or it may not occur at all. But it isn't logically necessary to explain (internal) entropy increase.
I personally like to think of all entropies as being "relational" entanglement entropies between subsystems. In this interpretation, the "internal" entropy is the entropy between two different spatial pieces that partition the entire universe. The "total" entropy can formally be thought of as the entropy between the universe and "everything else", although obviously if the "thing" is the universe then there are some obvious conceptual difficulties with exactly what we mean by "everything else". But the point is that entanglement builds up between the different pieces of the universe, but no entanglement ever "leaks out" to connect anything inside the universe with anything outside. (How could it? There's nothing outside the universe to get entangled with.)