Prevents arachidonic acid from being converted to thromboxane needed for platelet aggregation. Here's a diagram I find helpful of the arachidonic pathway and which drugs inhibit which parts of the pathway: https://files.mari.casa/aapathway.jpg
Linoleic acid can be converted by the body into arachidonic acid, so that part I got right all those years ago.
Inside platelets, chemical reactions are sped along by the enzymes. One of the important chemical reactions for clotting is the conversion of arachidonic acid (through a couple of intermediates) to thromboxane. Now, arachidonic acid is so simple: a linear, 20-carbon chain with a carboxylic acid group at one end, and four double bonds in the chain. Converting it to thromboxane takes a couple of steps: the enzymes need to kink the chain and oxidize specific carbons to make them polar.
The trouble is, aspirin inhibits the enzymes! So the enzymes inside the platelets can’t make their thromboxane. And, as such, clotting is inhibited, which results in bleeding. So this is why aspirin can cause bleeding