Think tanks don't lobby at all. What they do is design favorable conclusions for their backers, and the studies and press releases to dishonestly back those conclusions up and to generate pithy quotes. The politicians that are then lobbied by those backers now have something to say when interviewed about their political activities in favor of who is being lobbied for.

Politicians need a media strategy in order to do the things that they're paid to do by private interests, and think tanks provide that strategy.