Do lithium and beryllium want to lose electrons to be like helium?
Yes, lithium wants to lose electrons to be like helium because full valence shells are more stable states and all noble gases have full valence shells. So alkali metals lose one electron to achieve nearest noble gas configuration.

But beryllium is exceptional because of its small atomic and ionic radii. Due to this, its ionization potential is very high. So it mostly forms covalent bonds.