Is Jesus present only because the Lamp is lit?

No, the other way round: The lamp is used to remind that the Lord is present. (Though it should be lit always (Ex 27:20-21)).

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The tabernacle lamp has direct mention in various Marian apparitions, whether it is lit or not, whether someone would expect to find it but couldn't. Whenever I enter a Catholic Church that does not have a prominent tabernacle lamp lit it causes me a lot of concern, and I find it very hard to concentrate on my prayers. There was a prophesy that said the day would come when the tabernacle lamp's light would not be found in the Church -- as if to say that the Real Presence that is supposed to be there would be missing. It seems to me that day has arrived. Any Novus Ordo "Church" you enter more likely than not will give you the run-around to find the lamp. In the Mahony Square Garden so-called cathedral in L.A., any newcomer could easily spend 1/2 hour searching for the place where the tabernacle (and lamp) are hidden. Even having the information that it's located in its own room doesn't help too much. Once you enter the so-called room at first glance you wouldn't even recognize the entity they call the tabernacle as being a tabernacle. It appears to be a bronze column that partially survived a nuclear blast. Maybe that's the devastation of the Mass they're pointing at!