There are some small drawbacks associated with surgery, that all doctors are going to tell you. And there are some big drawbacks (like death or amputation risk), that not all doctors are going to tell you.
For instance, one of the greatest Brazilian female singers, Clara Nunes died some 30 years ago in one such operation, at the peak of her energy. Some people say this is old news, so here are more recent ones:
Woman dies after varicose veins surgery.
Woman has her leg amputated after varicose veins surgery.
Another woman dies because hospital failed to carry basic test.
Yet another woman died due to a blood clot in the lung following vein surgery.
Life-threatening infection almost costed woman her legs.
I should mention that probably the biggest risk lies with the anesthesia. Several other news relate people dying because of the anesthesia in other common surgeries, some of them purely aesthetic (like liposuction). This article is shocking:
(...) after decades of decline, the worldwide death rate during full
anesthesia is back on the rise, to about seven patients in every
million. And the number of deaths within a year after a general
anesthesia is frighteningly high: one in 20. In the over-65 age group,
it’s one in 10.
Other studies bring numbers way higher than seven in a million:
Brazilian and worldwide studies demonstrated a similar decline in
anesthesia-related mortality rates, which amounted to fewer than 1
death per 10,000 anesthetics in the past two decades. Perioperative
mortality rates also decreased during this period, with fewer than 20
deaths per 10,000 anesthetics in developed countries.
That's 1 death in 500 anesthetics. I'd prefer to keep my varicose veins, at least until I know the numbers about the risk they represent.
Also, when you look for information in doctor's and medical clinics websites, they don't explicitly mention all the risks involved. Here and here are two Brazilian examples. In English, we have other examples. The respected Johns Hopkins tells nothing about the biggest risks. Their "About" page clearly states their measure of success:
Johns Hopkins Medicine (...) is an $8 billion integrated global
health enterprise and one of the leading health care systems in the
United States.
This website puts it clearly:
Surprisingly perhaps most vein clinic websites fail to highlight
potential risks giving the impression that all treatments are
perfectly safe with nothing at all to worry about.
That's understandable: they want the patient's money, not to scare them away.
On the other hand, sometimes having varicose veins may also be risky, like this report suggests.
Unfortunately, I have found no studies comparing which risk (to keep or to remove varicose veins) is higher.
Finally, someone asked me about a source for my claim that medical error is now the third cause for death in the USA. Here it is:
Is US Health Really the Best in the World?
In case you don't want to pay US$30 just to have 24-hour access -- yes, they call it buying this sort of vital information -- you may try sci-hub instead. The relevant information is on (non-free) second page:
These total to 225 000 deaths per year from iatrogenic causes. (...)
[these estimates] are lower than those in the IOM report. (...) In
any case, 225 000 deaths per year constitutes the third leading cause
of death in the United States, after deaths from heart disease and
cancer.
Here you have some more information.