Yes, that is a theory. But in order to have been a star Jupiter’s mass would have had to be much more massive then it currently is. At least a hundred thousand times more massive.
Jupiter’s mass is 1/1047 that of the sun. If it were 100,000 times more massive it would be 100 times more massive than the sun, and would be a super giant star.
In order to be a star, to fuse hydrogen into helium, requires enormous pressure and heat. Fusion only occurs at the very center of the sun, heat is transferred by convection to the surface and radiated away. The specific power generation of the sun is about the same as a bale of rotting hay, about one half watt per cubic meter. The sun just does it longer - for billions of years - and with a lot more cubic meters.
Off by a few orders of magnitude. The minimum mass for a red dwarf is around 80 Jovian masses. That is the minimum to start the p-p fusion chain. For brown dwarves of 13 to 80 Jovian masses, one might find deuterium burning or lithium burning, but that does not last long. The energy currently given off by Jupiter is largely due to gravitational contraction and isotope decay.