I think it very much depends on the bicycle, your own body and the terrain/usage.
If you ride on flat terrain at a continuous, high speed then aerodynamic wheels will help you the most. Especially if the rest of your bicycle (+riding position) is already optimized. More gears will only help you if they have smaller gear steps so you can pick the perfect gear for the desired speed/cadence.
If you ride on hilly terrain then weight and gears both play a role. Though aerodynamics shouldn’t be underestimated. A wider gear range (especially easier gears) can make you ride longer and more efficiently.
If you are tall and/or heavy then most of the weight and aerodynamic drag will come from your own body and you’ll benefit less from better wheels.
A good set of wheels can easily cost 800€. The whole 2x10 Shimano Tiagra groupset (including crankset) is only ~390€, Shimano 105 or Sram Rival 2x11 is ~480€. If your bicycle is as old and heavy as the Claris groupset suggests then maybe you should consider a whole new (possibly used) bicycle. Though of course you can also start with the wheels and get a new frame+groupset in the future. The only consideration here is whether you’d like to have disc brakes in the future.