What's the current best method to remove stubborn spider/thread veins around the nose?

Sorry that you have been disappointed by the device treatments you have undergone. These machines tend to be far more backed by device manufacturer supported marketing hype than any hard science. For decades, light electrodessication has been successfully been used for treating telangiectasias and small veins on the face and around the nose (particularly those located immediately within the recesses behind the alar wings of the nose).  Two to four treatment session spaced at monthly intervals are typically needed to achieve optimal results. Make sure that you consult with a board certified aesthetic physician and insist upon seeing his/her personal before and after photos before agreeing to proceed. Best of luck. 

For varicose veins, we typically recommend a safe, minimally invasive procedure called Sclerotherapy. A salt solution is injected directly into each vein, causing them to collapse and ultimately disappear. For long-term improvement, we often complement Sclerotherapy with IPL (Intense Pulsed Light) photofacial and the Excel V laser, which treats both superficial and deep vascular conditions on all skin types. I suggest seeing an expert for a formal evaluation to determine if this is the best solution for you. Best, Dr. Emer

I treat nasal veins with a combination of sclerotherapy for the larger veins (large enough to insert a 33g needle) and microneedle RF for the smaller veins and, often times, I use a combination of both on the veins.  No treatment is permanent.

I prefer the Nd:YAG laser for veins around the nose area or any larger vessels on the face or body.  I get best results with this laser.  Off the face, my preferred treatment is injection sclerotherapy.  Nose veins can be stubborn and require more than one treatment.  Veins can also recur with time so accounting for some period maintenance is a good idea.