No. An unpredictable agent committing mass killings is a target, period. Before they knew his identity, they knew he killed according to his own agenda regardless of the law. As the pursuit went on, he killed with even less regard, taking out petty criminals just to maintain a hold over the populace. As well as taking out the government's own agents. No government would just "sit back and let him do his thing" no matter how opportunistic you wish to characterize it as.
Furthermore, at some point it would be established that Light is without a doubt the killer, as well as the method. A lot earlier than in the manga, too, as the "Just as planned" convolution and genre blindness started going wild around when he was finally put into confinement. And at that point, he'd simply be discarded. No agency on Earth desperately needs or wants a rogue, disgruntled, and unpredictable teenager to do their work for them.
I would have liked it if the series had been about what government(s) would do with the Death Note(s) once obtained. Though that would be more in involved LoGH political territory than the mind games setting of Death Note.