Is electrocautery the best option to remove prominent under-eye sebaceous glands? What are the risks?

There is no “best” treatment options. Cautery, lasers, high concentration peels may be equally used to destruct the gland. All methods have similar profile of side effects - pigmentation, scarring, infection. Blindness should not be an issue if precautions are taken. Lesions are very superficial, there is not way to destruct big vessels that are deeper in the skin. You should have realistic expectations, because they may reappear again (remember it’s your normal oily glands!). Sometimes medical treatments help (topical or systemic low dose retinoids) as an alternative. Good luck!