Does a balanced chemical equation tell you which reagent is in excess?
No, but it does establish stoichiometry.
A balanced chemical equation establishes stoichiometry. It tells you the molar quantities required for complete reaction. Typically one reagent will be excess, and one will be in deficiency. For combustion reactions, typically the oxidant will be in excess, i.e. if I had 16 grams of methane, which I burned in air, I could clearly designate methane as the reagent in deficiency:
This balanced equation tells me unequivocally that 16 g of methane requires 32 g of dioxygen for complete combustion. It will also tell me the quantity of heat I could get out of such a reaction given a few more parameters.
This is why chemistry teachers go to such pains to establish mass equivalence, and stoichiometry. A balanced chemical equation is a requirement to establish stoichiometry.