Yes, Tesla was a genius. Yes, he gave us many great inventions. The tesla coil is not one of his really 'great' ones.
It is a high frequency static electric charge generator and that's all. It can excite ions from a distance just like holding a fluorescent bulb under a HV transmission line will. That barely qualifies as work. Here's why.
Fluorescence works by big changes in static electricity (being brief here). The old and even the new CFLs that operate on this principle are much more efficient than an incandescent bulb. But not nearly as much as they could be. The ballasts in the old style and the electronics of the new, both use much more power than the actual bulb. Without this loss, they would be much more efficient.
How efficient? They would operate from a couple feet away from a HF source of low power or a few hundred feet from a HV source of high power. In short, this is replicating the actual mechanism use to light a CFL. It's just doing it centrally.
How efficient? It's not. Take the total lumens produced, divide by the lumens per watt and then divide that by the total power used by his coil and see what you get. My bet is it's around half the efficiency of one CFL. So, this is not groundbreaking, it's a scam on people's lack of awareness.
So, TL:DR version is, not more efficient. Tesla coils ability to send power suffers from inverse square law (power drops 4 time when distance doubles). This cannot power anything but exciting ions (CFL type stuff) without losing 99% of it's efficiency. It's existed for more than a century (Tesla did it continuously in his lab). You are subjecting yourself to magnitudes higher levels of EMI than so-called-dangerous "smart meters". It certainly won't be cheaper.
We now return you to your regularly scheduled programming.