The apparent advantage would be having a leader that they could see and touch. The Israelites, when one reads closely, did not exactly have a resounding faith in God,and it shows in at a few different points in the new testament, like in Exodus 14:10-12 where the Israelites were angry at Moses for leading them to the Red Sea when Pharaoh's army approached, or in Exodus 16:3 where they grumbled at Moses because they had no food. It is also demonstrated in their lack of fear for God when they rebelled so many times in Judges. When you break down the reasoning that hey had in 1 Samuel 8:20, you can see that hey want someone to fight their battles for them, which means that hey didn't see God as capable of doing so, despite the fact that he had done so already many times over.
There is also the first reason, which comes up twice, that hey want to be like other nations. The Israelites were God's chosen people, but hey had the heart of Esau, which the author of Hebrews describes in 12:16
"See to it that nobody is sexually immoral, or Godless like Esau, who
for a single meal sold his inheritance rights as the oldest son"
They didn't much care that hey were set apart from other nations by having a literal god leading their nation, they wanted more desperately to be led by a man and fit in than to be led by a God and stand out, so they sacrificed one aspect of their holiness for it.