I know it's not 100% but does EVLT really work for spider veins?

if the only complaint you have regarding your legs a spider veins, then you most likely don’t need an EVLT. Unless you have swelling, aching, bulging varicose veins, or pain in your leg, you most likely do not have anything to benefit from having the EVLT procedure. There are unfortunately a lot of places that do EVLT on everyone who walks in the door even if they don’t need it. I have seen a very large number of patients who have spider veins and that is their only complaint. They have been elsewhere and told they need 1 to 6 EVLT procedures to treat their spider veins. Once I repeated their alter sound I found that they did not need any EVLT procedures and they just need it to have their spider veins treated. I would seek a second opinion before I had an EVLT in the absence of any of the above. 

Dilated painful varicosities and smaller spider veins are often the results of significant GSV reflux.  If you, your doctor and the insurance agree that you are having significant symptoms from your GSV reflux, then they may pay for treatment to correct this.  These treatments will stop or prevent spider veins. If they are already present, consider sclerotherapy or laser treatement for them.  Since they are usually cosmetic (unless there is a wound or impending wound) their treatment is done out of pocket. 

Spider and varicose veins are often but not always associated with reflux in larger veins.  A proper evaluation and ultrasound can help determine this.  If there is in fact reflux, treating the large veins with EVLT makes treatment of the spider and varicose veins more successful and makes it less likely that you will develop more of them overtime.  It's important to treat the source of the problem.

EVLT is pushed by various doctors for a variety of reasons, most of them evidence-based and life changing, some of them dubious.
This falls into the dubious category. If your doctor is telling you that you should get treated (and your insurance will pay for) to avoid spider veins, run - don't walk - to get a second opinion.

To answer your question, it may depend on what is "mild" venous insufficiency. The accepted standard of "abnormal" is a valve closure time of greater than 0.5 sec.  If you have mild reflux and no leg symptoms, vein closure is not indicated. There is a general feeling that treating reflux decreases varicose veins in the future but limited studies.  Red spider veins and blue reticular veins can occur in the absence of any refluxing vein valves.