Plus all of the infrastructure to tie into the grid is already in place. Seems like there's a lot of benefits.

In principle, sure. Lake Mead is in a warm enough location that it never freezes over (which could pose some difficulty for other lake-floated solar) and the intense heat and low humidity mean that quelling evaporation could be very beneficial.

In practice, the area of Lake Mead is about 250 square miles = ~160k acres * (~200 kW/acre) = ~30 GW of potential peak generating capacity. This means that a solar farm covering Lake Mead would immediately become the largest power plant in the world at peak output, surpassing the Three Gorges Dam (hydro) by almost 50%. It would be more than a quarter of all solar capacity existing in the United States as of 2022. So the overall electricity sector may not be ready for such an undertaking.

Partial cover is more feasible at this point, eventually extending to the entire lake, hopefully.

correct. It's generally not feasible to shade such a large area over water. But evaporation does take its toll on reservoir water. There was a proposal not long ago to shade the canal carrying water from Northern California to Southern California. Reducing evaporation was one of the touted benefits[1].

  A study by the University of California, Merced gives a boost to the idea, estimating that 63 billion gallons of 
  water could be saved by covering California’s 4,000 miles of canals with solar panels that could also generate 13 
  gigawatts of power. 

1 - https://www.pbs.org/newshour/economy/solar-panels-built-over...

Not even directly that I think, it's to prevent chemicals that are used to stop algae growth to interact with sun.

Shading is used in California to prevent evaporation, see for example the well-documented use of "shade balls": https://en.wikipedia.org/wiki/Shade_balls

Their use in Los Angeles was covered in a Veritasium video that was all over YouTube for a while (97M views): https://www.youtube.com/watch?v=uxPdPpi5W4o

The estimate given when they were added in 2015 – in the middle of a record drought – was that they would prevent the loss of 300 million gallons of water each year.