Any plastic surgeon or dermatologist experienced with many lasers (not just the one or two other doctors may have) can successfully eliminate most of these facial telangiectasias with a single treatment session with a KTP laser. (I always offer a free touch-up session no sooner than one month after the initial treatment session to ensure maximum improvement.)
As a general rule, you should probably avoid any doctor who has a "one-laser-fits-all" attitude, as most lasers have specific uses based on wavelength, power, and shutter duration. It should NOT be "Well, I have a hammer, so everything must be a nail!" Even, if as in your case, you have had two different "hammers" used. You need the laser best tailored for the problem at hand!
Treatment with a KTP laser may cause superficial blistering and/or crusting for a few days, but this wavelength laser (532nm) is very well absorbed by blood and skin pigment, making this a workhorse laser for tiny blood vessels and superficial pigment spots. Scarring is rare with proper treatment, and once the vessels are gone, they remain gone.
KTP stands for Potassium (K) Titanyl (T) Phosphate (P), which is the frequency-doubling (wavelength-halving) crystal used in the beam path of a 1064nm near-infrared Neodymium:YAG (Yttrium Aluminum Garnet) laser. This was the first of many lasers I purchased over 20 years ago, and I still use it weekly for problems exactly like yours.
The V-beam is a 585nm yellow light laser that is also well-absorbed by blood, but this laser is a flashlamp-pumped pulsed dye laser with long(er) shutter durations than "standard" 585nm pulsed dye lasers, which also emit laser energy at 585nm, but in very high-energy, short shutter duration (450 microseconds, about half a millisecond) pulses. Pulsed dye lasers were designed to treat the extremely small surface capillaries in port wine stains (capillary malformations), and this type of laser has too short a shutter time for the thermal relaxation time of larger diameter vessels (which are your [small] facial telangiectasias). The V-beam laser would be used in the 10 millisecond pulse duration range, which may still be a bit too short of shutter duration (inadequate energy) to eliminate vessels the size of your facial telangiectasias.
A V-beam laser is not a "bad" choice, but is probably not be the "best" choice. Other less-than-ideal options that can (but don't always) work include IPL (non-laser Intense Pulsed Light machines), argon laser (ancient blue-green laser with too long pulse duration and too little energy = non-specific for blood vessels), or other lasers that are NOT recommended.
There are probably several physicians in your area that routinely use a KTP laser; seek one out and make sure the operator is experienced in its use. Best wishes!