The inability to execute without Roman approval had nothing to do with citizenship.

The rights to capital punishment were taken away over a period in which Jesus was killed.

But the accounts of Jesus's death are extremely unusual given the other reports of messianic upstarts in Josephus who were killed by Roman forces without trial, immediately upon gathering, and where followers were killed too.

The allegation of it being at the Sanhedrin's urging would be extremely unusual if true. And Roman reluctance even more so.

One might even look to alternate charges that were publishable by death under Jewish law but accepted in Roman society for the kind of charge that might lead to such an outcome.

Such as the charge of homosexuality. So if there were reports of kissing or feeding a close male disciple food at dinner right around when Jesus is arrested, we might want to entertain the possibility a historical Jesus was killed by his own people for allegations related to that, which wouldn't have been an easy decision for a Roman authority given the rumours even the Emperor at the time was engaging in some behaviors.

Another might be rejection of intelligent design in favor of Roman philosophy, like Leucretius's "seeds of things" scattered randomly where only what survived reproduced, and the seed that fell by the wayside of the path did not. Those are all Leucretius's words, and yet it sounds very similar to a saying by Jesus which is offered up a secret explanation for its public utterance in canon. Whereas in the tradition of the document I think with greater connection to the historical origin, they believed that parable was about indivisible points which make up all things and were the originating cause of the universe (their words).

That document says things like:

Jesus said, "If the flesh came into being because of spirit, that is a marvel, but if spirit came into being because of the body, that is a marvel of marvels."

In fact, the saying immediately before the parable in this work was:

The person is like a wise fisherman who cast his net into the sea and drew it up from the sea full of little fish. Among them the wise fisherman discovered a fine large fish. He threw all the little fish back into the sea, and easily chose the large fish. Anyone here with two good ears had better listen!

So a tradition of Jesus that was engaged with the ideas in Leucretius's De Rerum Natura - the only extant work from antiquity to explicitly describe survival of the fittest - might also be a tradition that was deemed by the Sanhedrin to be 'heretical' but not one so easily dismissed by Roman authority in a time Leucretius's book was still quite popular across Rome.

It would be incredibly unusual for a made up tradition to have also made up a schism where their devout Jewish messiah figure was paraphrasing the brightest Roman mind on what was wildly transgressive at the time yet since proved to have been true. Or to have added in intimate moments with his supposed betrayer and public denials by his supposed successor.

But these are exactly the kind of details we might expect from a version of events contending with living witnesses of actual events that need to be addressed and spun in a different way.

It's a bit like Perseus and Medusa. The only way to spot what was really going on is in the reflection left behind by its opposition in the writings of the victors. But that reflection can actually reveal quite a lot.