Why can't $rm N_2$ TEA lasers produce optical breakdown? (sparks in the air)

The explanation is simpler than you think.

Just like CO2 lasers do not produce a visual breakdown of atmospheric air (or most other IR lasers), TEA lasers operate in the ultraviolet range: 337.1 nm.

This answers in theory why weak lasers would not breakdown air, but why don't strong lasers break down air?

This is because of limitations of electrical properties of the materials used in construction of the laser. There are major two factors that impact the power of a TEA laser:

• The atmospheric makeup of the resonator cavity
• The dE/dt of the resonator cavity

Paradoxically, the same factors that would increase the power of a TEA laser via changing the atmospheric makeup also force a decrease in dE/dt. This limits the practical power for any given length of laser cavity, and thus many TEA lasers could not achieve atmospheric breakdown under perfect conditions.

This is in combination with the fact that the coherence of TEA lasers is, franky, doodoo, and with all other factors, it comes down to the simple fact:

The power per area of TEA laser beam does not exceed the requirement to break down air, it is possible, but the construction required may be hundreds of feet in length to garner enough power, and that is impractical to put into a light N2 vacuum.