Interesting question! Yes, the hydrophobic amino acids are very important; they facilitate interaction with the hydrophobic (inner) portion of the lipid bilayer. A useful review focusing on the bilayer side that I'll reference throughout this answer is here: Mechanics of membrane fusion, doi: 10.1038/nsmb.1455.
(a) shows a model of two lipid bilayers fusing (minus protein contributions). Essentially, the spike makes this process thermodynamically favorable.
The hydrophobic residues are key because they are what enables the other end of the spike protein to interact with and deform the cell membrane. Here's a model of protein-mediated fusion from that same review (they emphasize that the deformation is key):
The cleavage of spike is crucial to initiate this process because the hydrophobic residues are not accessible/cannot interact with the cell membrane unless the spike has been cleaved. This works to the advantage of the virus, since if the spike is close enough to be cleaved by TMPRSS2, it's close enough to stab the membrane of the correct cell. A virus with hydrophobic residues that are always ready-to-go, in comparison, would likely fuse with a lot of random/off-target membranes of cells or vesicles that would not be permissive to viral replication.
Mechanisms of membrane fusion disparate players and common principles is a review paper.
A detailed discussion of the physics of this process is beyond the scope of this site, but here are some potential starting points (disclaimer: I am not a physicist):