If a YouTube video I saw whether you can use moonlight to start a fire is to be believed, you cannot achieve higher temperature using lens than the source temperature, so your earth lens will achieve same temperature as 5cm lens from dollar store.
With a mirror you can also burn things if you focus sunlight, but the surface of the mirror stays cool.
An atmospheric lens, however, will reach a maximum of somewhat closer to solar surface temperature, though still lower because of scattering and absorption which definitely isn't trivial on this kind of scale.
https://www.universetoday.com/19981/moon-albedo/
If it had an albedo of 0% it would be a black body and be radiating only thermal radiation at its surface temperature; it's not too far away from that but it is reflecting some "black body" radiation from the sun representative of a higher temperature.