I think there are more important things than fairness. We have laws requiring employers to give you paid time off for jury duty and it's still a significant burden on a lot of people. For someone to be chosen by lot to participate in what's likely to be the equivalent of a full-time job? Frankly, it would be unfair to place that burden on people who didn't sign up for it.
And sortition would be pretty much guaranteed to fail if the people chosen by lot weren't drawn from a pool who'd already proven they understood, to some degree, the issues they would be making decisions about. If we take conscripts instead of volunteers, we'd have a committee full of people who didn't know, or care, about the issues, who didn't want to be there in the first place, and who'd happily rubber stamp the decisions of whoever on the committee was the loudest and most passionate so that they could go home early.
And I think you have to draw a distinction between influence and corruption. Everybody has the right to advocate for policies that benefit them. Everybody has the right to form unions and groups and lobby for policies that benefit them. So of course this committee will be influenced by outside parties. The alternative is decision makers who don't listen to people telling them what they want and need, and that's not good for society.
And of course there will be explicit and implicit bribes, there will be be influence campaigns, and so on, because you can't have any decision making process, anywhere, that people won't try to rig.
But: it's much, much more difficult for some corrupt billionaire to bribe 100-plus members of a committee - especially if, as I hope, that committee makes decisions by consensus, anarchist style, instead of by majority vote - than it is to bribe one politician. And once you've bribed all those committee members, as soon as their term expires you have to bribe a whole new committee, because there's no guarantee that any committee member will be selected by lot next year (and, in fact, I'd argue someone shouldn't be eligible to serve again for x number of years after being chosen). On the other hand, bribing a career politician is a continuing investment that pays off year after year after year.