Exactly. If anything, we need to transition to a democratic model based at least in part on sortition. Randomly selected citizens assemblies have already shown promise in Ireland, France, and elsewhere.
So a purely random selection would seem like an improvement at this point.
Just as politicians may not be experts in the specific areas that they legislate on, the members of this second chamber would have the right to ask for input from specialists. This could be seen as an extension of the right that juries have in some states to ask questions of expert witnesses.
Even if, in practice, the second chamber end up mostly listening to advisers (or lobbyists) provided by the existing political parties, they would at least have to hear a variety of opinions, let's say in proportion to the partisan leanings of that chamber. This would eliminate gerrymandering and other problems like campaign financing.
The deliberative structure used is based on research into group decision-making, and is usually some form of system where people split into small discussion groups that engage in structured discussion, then present something to the broader group, then discuss again, then shuffle into different groups and repeat the process.
The fact that people are self-interested also doesn't pose a problem, so long as the sample is large enough. You probably want at least 100 for good measure, depending on the population being represented. Because it's a representative random sample (ideally it's not fully random, but weighted to represent population demographics), you get a broad slice of the population and therefore many competing interests. An individual may wish to further their own interests, but this would conflict with the interests of everyone else. Overall, the group's incentive is to do a good job, because they live in society and their decisions will affect them.