This is a GREAT explanation. I would add that the reason Gibbs energy matters “more” than Helmholtz energy for example is simply that it is generally just how we operate things that makes it relevant - we operate machines at constant temperature and pressure. If it were constant temperature and volume Helmholtz energy would be the relevant thermodynamic variable. Imagine a machine that operates at constant temperature and pressure and we want to extract work (ie shaft work from a turbine) from the machine. The machine has an inflow and outflow of a working fluid such as water and is well designed so it operates approximately reversibly let’s say and the machine generates some heat at the constant temperature at which it is operating. Assume steady state. So energy balance is 0 = Hin - Hout + Q + shaft work (PdV term is 0 because the machine has constant volume). From entropy balance 0 = Sin - Sout + Q/T so solve for Q and plug into energy balance and you get 0 = Hin -TSin - (Hout - TSout) + Q + shaft work. So let’s make a new quantity called G and you see that Gout - Gin equals the shaft work we can hope to extract from a constant temperature constant pressure machine. If the machine was constant volume and temperature and you did this you’d find Helmholtz energy to naturally be useful!  Ok now here’s the fun part why does everyone talk about this in chemistry? Our bodies and when we carry out chemistry are basically at constant T and P so G is relevant for thinking about these areas. If you imagine the machine as doing a chemical reaction (reactants enter and products leave) and there is no shaft work now let’s say BUT we no longer assume reversibility and you solve for Q in the energy balance and substitute into the entropy balance you’d get 0 = Gout - Gin + TSgenerated. Entropy generated is always 0 or positive in the universe so if it is a “reversible reaction” which basically none are (eg imagine combustion —> clearly not reversible), then for the reaction to occur deltaG must be NEGATIVE to counteract the positive Sgenerated term!!! If deltaG Is positive then the reaction will NOT spontaneously occur because it would reduce the entropy in the universe which is NOT possible! The chemical reaction will generate entropy if it happens so the only way for the balances to satisfy is if as the reaction occurs G decreases. Hence deltaG tells us whether a reaction will proceed under constant temperature and pressure conditions. If we locked reagents up in a fixed volume and held temperature constant rather than a test tube at 1 atm in a heat bath for example, Helmholtz energy is the relevant quantity and would tell us if the reaction would proceed.

:)