First, all arguments against actual infinities commit the fallacy of applying rules of finite arithmetic (which are only logically valid for finite quantities) to transfinite quantities. All professional mathematicians agree there is nothing illogical or impossible about an actual infinity, and in fact all logic and mathematics depend on the fundamental axiom of infinite extension being true. For example, there must be a set that contains all numbers simultaneously, and that is an actually infinite quantity; and in the case of theism, all numbers must exist in the mind of God, therefore, either way, an actual infinity exists, and if it can exist of numbers, it can exist of anything, like seconds--to which one could assign every number, so if an infinite quantity of seconds can't exist, then neither can all numbers exist, yet obviously it is absurd to say some numbers exist and some don't; since all numbers exist, there can be as many seconds as numbers, one for each number, therefore time can be infinite.
Second, all arguments against a past infinity of time commit the fallacy of circular reasoning, e.g. "if the timeline began infinitely long ago, we would have reached this point in time infinitely long ago" presumes that the timeline "began" which is exactly what isn't the case if time is past-infinite. If time is past-infinite, then all points in time exist, which means our point in time exists, too. So there is no difficulty in our being here. Just as there is no difficulty in seeing that the number "minus five" exists even when there are infinitely many numbers before it on the numberline, so there is no difficulty in seeing that the time "now" exists even when there are infinitely many seconds before it on the timeline.
Third, the argument that the Big Bang had to have been the beginning of time has no scientific basis. Just because the presently observable universe began 14 billion years ago (with our Big Bang) does not entail there was no time before that moment. In fact, as Stephen Hawking himself conceded, the Hawking-Penrose theorem that once proved time began with the Big Bang, is false. Modern physics entails that singularities are impossible, therefore there could well have been a time before ours, which we just can't see because our Big Bang erased all evidence of what preceded it. In fact, there could have been an infinitely many Big Bangs in the universe's past, ours just being one in the chain, one of the few that generated states of the universe that could produce life, which is why we are here instead of elsewhere in the chain. In fact if there have been infinitely many past Big Bangs, even after subtracting all the ones that couldn't produce life, there still would have been infinitely many past universes with other people in them much like us. We're just one of those.
In fact, according to quantum mechanics (one of the most confirmed theories in all of science), any Big Bang, of any initial size and entropy, has some probability of occurring, anywhere, at any time. But that probability is extremely small--so small that we can expect this universe to decay entirely into a black amorphous haze of nearly flat radio waves, a maximum entropy heat death, after millions of billions of trillions of years, long before another Big Bang will occur from a random fluctuation in that haze. But since this universe will continue forever, eventually any improbable event will become inevitable (since anything that has a nonzero frequency in an infinite set necessarily occurs in that set, no matter how small that frequency is--it's just a matter of how long it takes), therefore it is 100% certain that another Big Bang will occur, when our universe has almost completely decayed away, and if there come to exist any people on the other side of that Big Bang, they will have no idea our universe preceded it, because what little evidence there was left will have been erased by that Bang. We might be those people, looking back at a Big Bang that occurred at the long end of someone else's prior decayed universe. And this may extend back through all infinite time, one Big Bang after another, without end, just as will be the case in the future (since there is nothing that will stop this endless cycle of decaying universes and inevitable Big Bangs).
Even if (as some physicists now conclude) the accelerating expansion of our universe causes it to tear apart, the energy released from that universal Big Rip will in fact cause a massive, universe-wide Big Bang (or possibly countless many Big Bangs, which would rapidly separate from each other and thus create many universes simultaneously that can't see each other). And even if the universe unexpectedly slowed down and collapsed in on itself, that, too, would re-ignite yet another Big Bang. Therefore there is no way to stop an infinite future series of Big Bangs. And if there will be an infinite future series of Big Bangs, there could be an infinite past series of Big Bangs, and very well may have been.
And since this follows necessarily from known physics, we require no additional assumptions (like a marvelously complex and convenient super-god) to explain why we are here: since there are infinitely many Big Bangs (and this follows necessarily from the confirmed facts of quantum mechanics), that one of them would generate a universe tuned to produce life by accident, is statistically inevitable, and thus guaranteed. Thus if there was an infinite past time, we would necessarily exist (in fact, this still follows even if there wasn't an infinite past time: even if it all started with a single random Big Bang, we could still be at the end of a long ensuing sequence of inevitable Big Bangs, the first one to randomly get the mix right to produce life). Therefore we do not require any design hypothesis to explain how we came to exist.
One might still ask of course why there is an infinite timeline, but that is no more cogent than asking why there is an infinite god. "Time" we at least can prove conclusively exists. God, not so much. And if an infinite god can exist for no reason, so can infinite time. Yet infinite time (or even a very long but finite time preceding our Big Bang) is a vastly simpler cause than a god, having only two attributes: temporality and extension (i.e. a single, simple dimension). God has many more attributes than that, and therefore it violates Occham's Razor to prefer God to "very long timeline" when the latter fully explains all we observe.