It can also be explained by branching. Run a simulation of a universe, and every time a random bit is required, fork into two separate processes. One where the "random" bit is 1 and another where the bit is 0. From the inside, it would seem indistinguishable from true randomness. But the system as a whole is purely deterministic.
When I was younger and had most interest in non-useful things (in a boring sense) I came to "the theory" that each time observation or logical induction is performed, the universe splits into many variants ahead of time and these variants that produce "oops" in terms of contradiction simply disappear. Each time you see something weird but real, it is oops that survived, because you didn't see something contrary yet (and now you cannot, because only one thing has to be remaining).
That made me sad, because we could come to hyperdrives and FTL journeys to alien worlds, but some people in 20th century made few observations and conclusions that now prevented fun forever. Magic things were easy before the technology, now they are physics-hard. Our universe is spoiled in a very wrong way.
How would forking occur? Forking requires copying data. Does the universe get cloned infinitely many copies all the time? That's quite an extraordinary claim.
This answer seems to be assuming the universe must be implemented on a classical computer and computed in a reasonable amount of time (that's not slowing down exponentially over time). That's not necessarily true.
I feel that assuming that the nature of how the universe is computed matches up with our first intuition feels a bit like the assumption in geocentrism that the universe matches up with our intuition of Earth being the center and the most significant body.
Even if you accept the basic premise of the universe being a simulation, there are still many ways it could work with MWI. The simulator's universe needn't have classical physics; the simulation could be running on quantum computers or something more advanced. Or the simulator's universe could be classical with a classical computer doing the simulation at exponentially-decreasing speeds as it has to simulate the increasing number of branches. It wouldn't make any difference to us how long the simulation takes to compute us, as long as the simulating machine doesn't break down and succumb to entropy. The simulator's universe could be something like Conway's game of life, and the simulation is running on a turing machine pattern which will never decay or break down. (I might be borrowing more than a few ideas from Permutation City here.)
Note that you're assuming that a nanosecond is a tiny amount of time. It's tiny relative to what we're used to, but as far as what "implements the universe" goes, a nanosecond could be enough time for a huge amount of occurrences to happen within. We don't know.
Amazingly, we have an experiment that shows that this is not the case, if you make some pretty reasonable assumptions. It's based on Bell's theorem. Your statement is very intuitive, and the fact that it may be wrong is very surprising.
Not everyone agrees on what is "reasonable". I have no problem giving up locality, but "true" randomness (i.e., information generated without an algorithm) seems like a philosophical cop out.
Note that something can also be globally deterministic but indeterministic from the perspective of a subsystem. In this sense, the universe would appear random as far as we're concerned, but not random to whoever is simulating the universe (this is often called superdeterminism, but I think that's a silly word—the universe is either deterministic or it isn't).
The probability of an event depends on your knowledge. Whether something is deterministic is a question of probability. Hence whether something is deterministic can depend on your knowledge. If the universe is deterministic to its simulator but random to us, I would call it random. Besides this difference in terminology, I think we agree.
Do you agree that the probability of an event depends on your knowledge? For example, what is the probability that the bottom card of a shuffled deck is an ace? Peek at the top card, it's an ace of spades, now what is the probability?
Would you agree that something is determined iff its probability is zero or one? I'm guessing the disagreement is here - what definition would you use?
One concerns a lack of knowledge, like in your cards example. In that case, the probability is an expression of your state of understanding of the deck, and is not a property of the deck itself. The physical details of the deck and the situation it is part of could be entirely deterministic and we could still talk about this kind of probability. In this case your knowledge is independent of whether it is probabalistic or not.
The other is whether the universe contains fundamental randomness, such that you could say it is literally "probabalistic". And in this case whether that is true or not is independent of our knowledge of the probabilities.
You are right, according to quantum mechanics. If you have a pair of electrons in a product state and measure the spin of one of them, you instantly know what the "other particle's" spin is going to be (if you knew what the initial state was). When you "peek at a card", you "make a measurement" and alter the system.
For anything that exists outside of a light-cone around the first electron, a measurement of the second electron will be truly random from the measurer's perspective.