If he had done the exchange at the very begin of the manga, it would have helped him in a sense of the FBI stalker, but not towards killing L as he doesn't show up until late stages.

Light and L also cross-paths a few times however Light doesn't know his real name or face, just yet.

L also left to Watari that is anything were to happen to him, that the suspicious raised towards Light would be more than just probabilities.

So in short until Light is already caught on L's trap there was nothing that knowing his name would help with and by that time Light already had Misa by his side so there was no point to it.

Also keep in mind the Ryuk can see his lifespan and he probably already knew Light would die some time soon.

Hypothetically speaking(since that is what it means to give Light the shinigami eyes at an early stage), if you give Light the shinigami eyes, the early he could have killed L, was when they were frequenting the same school/university and at that point L was already highly suspicious of Light to the point of tracking him.

If L were to die there regardless of how, it would trigger L's successors to come in with all the already collected information which again would be focused towards Light with the prediction of not to expose yourself to Light.

There is the fact pointed that Light could not kill without knowing one's name based on L's experiment, but would that be suffice for them to cease investigations at such an early stage which not even L ceased him self?