I haven't spent much time defending the LCA, but it's not because I think it's a weak argument. As my readers know, the vast amount of time defending theism proper I spend vigorously defending St. Thomas Aquinas' Five Ways. With that said, I consider the LCA, the conceptualist argument, and the argument from desire to be the next strongest. Let's take a look at the LCA:
1. Everything that exists has an explanation of its existence, either in the necessity of its own nature or in an external cause. (Premise)
2. The universe exists. (Premise)
3. Hence, the universe has an explanation of its existence. (From 1 and 2)
4. If the universe has an explanation of its existence, that explanation is a timeless, changeless, immaterial, powerful cause. (Premise)
5. Therefore, a timeless, changeless, immaterial, powerful cause exists. (From 3 and 4)
Premise (2) is hardly controversial. Premise (1) is known through experience. If there were an elephant standing in the middle of the street, and someone claimed that the elephant has no explanation whatsoever, then surely people would think he is either crazy or merely jesting! Nobody would take such a claim seriously.
Since the remaining premises, except (4), are equally uncontroversial, let's focus on premise (4). Would it really be reasonable to think the universe exists necessarily? Does every quark exist necessarily? Moreover, to exist necessarily is to have its existence and essence identical. Yet, the universe has diverse essences, which makes a necessary universe impossible.