It can be a symbol as far as shedding, but only his blood could be a sacrifice, no other blood. The fig tree is a symbol of the condition of Israel as a whole, and their judgment to come. Adam and Eve tried to cover their sin with fig leaves, a symbol of man covering their sin by their own works.
figs have.. not very big leaves. not the first thing you're going to go for if you're trying to make a skirt and you have every kind of tree available.
so, why figs?
they hid because of their shame, but the girdles weren't necessarily an effort to hide that shame - otherwise why also hide themselves?
if the fruit was the fig, then they covered themselves with the emblem of their shame
if Adam found Woman at the tree, and they then together did this, then the leaves were right there, and they both knew she had been deceived.
they didn't hide until they heard God approaching, and feared - and when they spoke to Him, they were repentant and confessed their sin.
i think one reason they didn't eat from the tree of life is because they knew they didn't deserve life, and another is that Adam at least, perhaps both of them, were wise enough to know that eating from it in their sin-dead condition would result in judgement. a bad idea. God kept them from it for a reason, for their good, until Messiah should come - - we cannot approach God without His propitiation, or we will die - just as the priest in the temple could not enter without washings and cleansing with blood, and could not be unclean in any way.