I'll have a go at this, from a perspective devoid of any biology (which I know nothing about). You want to wind the time back to some earlier "instant" (when life emerges on Earth) and then run the world from that "instant" on a second time around. I'm taking all that quite literally.
First remark. I put "instant" between thingies because it may be troubling from a general-relativistic perspective. Never mind that.
Second remark. You can't do that. You cannot wind the time back (and start over). Therefore, this is not a question of the normal scientific variety. Never mind that.
According to the laws of physics (as they are currently thought of; and those are freakishly accurate on our scales), if there is (objective/real/fundamental) "randomness" in the world then it can only be of the quantum-mechanical kind. Quantum mechanics typically deals with the very small stuff. (Ignoring Big Bangs and black holes.) Typically no-larger-than-atom stuff. Unless there is some kind of "natural" amplification mechanism at hand,* I don't see how quantum-mechanical randomness (if it indeed exists; there are other hypotheses and interpretations of QM around) with any significant probability can reroute a relatively macroscopic phenomenon such as evolution.
However, if quantum-mechanical randomness exists (e.g., no hidden variables, no superdeterminism, etc.), then (in my tentative** opinion) the probability that it will reroute evolution is not zero. Just very small.
I guess*** that at least 99.99% of all reruns yield exactly the same biological diversity that we have today. (And this will not change unless perhaps we start using quantum-mechanical amplifiers such as Geiger counters for our decisions regarding species demolition.) And if evolution was to be rerouted during the second run, it would most likely be immediately after the start of the second run, when life was barely existing and thus most vulnerable to quantum-mechanical mishap.
*I can only think of some intermediate chaotic process, but even that doesn't do it for me. (It might change the weather a bit, but evolutionary history? Nah, very unlikely.)
**The tentativeness is due to personal uncertainty regarding the causes of DNA errors. However, I now believe them to be (by far) mostly chemically describable in nature (especially those that produce offspring), not only quantum-mechanically describable.
***An actual computation of a good lower bound is impossible. I think I'm on the safe side.
NB: To those who "know" evolution to be inherently random, I would ask: Could you distinguish between evolution based on "true" randomness and evolution based on pseudorandomness? I would suggest that for all biological-theoretical purposes they would be equivalent.
Evolution (assumed true) doesn't come close to proving (or even suggesting) that the world/universe has inherent "true" randomness. Evolution doesn't require "true" randomness. It is perfectly fine (from a consistency-perspective) to accept evolution and be agnostic about "true" randomness.
It is precisely for the purpose of this question (with its "inevitable"), where the distinction between random and not-random matters. The “normal” assumption of randomness is very well suited to answering biological questions, but this question explicitly (and apparently inadvertently) wanders into another domain of inquiry: philosophy.
The OP's question asks about "inevitable". It is precisely in that word only, where the distinction between randomness and, say, pseudorandomness is crucial. If reality is pseudorandom then things are inevitable. If reality is random then things might be less inevitable. So, for the purpose of this question, we need to acknowledge that we really don't know about this aspect of reality (and usually we wouldn't care, because for normal questions it doesn't matter one iota).