Let's start with a simpler issue: Florida law may not require malpractice insurance but it puts extremely strict rules on doctors who do not carry it.
This website claims that Florida doctors who do not carry insurance must put some amount of their own money in an escrow account (as in, an account that can only be tapped under specified circumstances) to cover malpractice damages and prominently place a sign saying they are not insured. So even if Sneed really doesn't carry malpractice insurance (which I could not verify either), she is still required to have some ability to pay out if she loses a malpractice suit. If she doesn't have insurance
or follow the rules for uninsured doctors she's risking investigation by the medical board or Attorney General, especially since social conservatives won all statewide offices in Florida by double-digit margins earlier this month. There are plenty of people in Florida government now who are willing and able to make an example out of SRS quack butchers.
Now for your actual question: Not carrying malpractice insurance does not make Sneed immune from malpractice suits or change the burden of proof on the plaintiff. What it does is make it much more difficult for a successful plaintiff to recover damages. With malpractice insurance the insurer commits to covering any settlement or judgment, so if Sneed had insurance and lost a malpractice suit getting the payout would be as simple as sending the insurer a copy of the court ruling. If Sneed does
not have malpractice insurance and told the plaintiff "fuck you, not paying" after losing a malpractice suit then recovering money would become much more difficult: The plaintiff would have to go back to court to (1) prove Sneed has wages that can be garnished or assets that can be seized to satisfy the judgment and (2) prove they are
not protected by Florida's very defendant-friendly laws surrounding garnishment and seizure. This is all compounded by the extremely high cost of legal representation ($100+/hour or a 4-figure flat fee is a typical starting amount unless the attorney agrees to waive up-front payment in exchange for a cut of any payout) even in trivial cases.
To your question of "well why don't all doctors refuse to carry insurance then?" it's because most hospitals require their doctors to carry malpractice insurance (not an attorney but it's probably to minimize the hospital's financial exposure from malpractice suits) and because a doctor who doesn't carry malpractice insurance and refuses to pay out will be in extreme legal risk for life - just structuring things to avoid garnishment or asset seizure would probably be a full-time job.