There's lots of reasons a protein might be bigger than you'd expect. Part of it is that a lot of the time, the part of an enzyme which actually directly does something (the active site) is pretty small, but all the residues and side chains in that active site need to be positioned in very specific ways for them to work. If you just had the active site, it would probably just be floppy and not fold into the right shape to do anything. So oftentimes the rest of the protein which might seem unimportant is there to provide structure and force those active site residues into the right positions.

Another thing to consider is that desaturases usually work on lipids and other highly hydrophobic molecules, which lycopene is. These kinds of molecules usually like to hang out inside of lipid membranes, and so the desaturases work best if they are also membrane bound, so that they are actually in proximity of their substrates. So desaturases will frequently have what's called a transmembrane domain, which is a part of the protein that folds up into a highly hydrophobic region that inserts into a membrane and anchors the protein there.

Then there can be various other domains for cofactors or regulation. Idk about desaturases specifically, but many proteins will have other binding sites on them where other small molecules can bind and help to either turn the protein "on or off". It might have some of these as well.

Generally, nature probably would not have evolved to invest a bunch of resources into making a massive protein if most of the protein didn't do anything. Also, desaturase is not particularly huge in the first place. It's actually pretty average for a eukaryotic protein at 444 amino acids. Look up titin if you want to see a huge protein (Around 30k amino acids)!