Does the theory that "sincerely" originated from "without wax" hold any merit?

The adverb is from the 1530s:

sincerely (adv.):

1530s, "correctly;" 1550s, "honestly," from sincere + -ly (2). As a subscription to letters, recorded from 1702.

sincere (adj.):

sincere (adj.) Look up sincere at Dictionary.com 1530s, "pure, unmixed," from Middle French sincere (16c.), from Latin sincerus, of things, "whole, clean, pure, uninjured, unmixed," figuratively "sound, genuine, pure, true, candid, truthful," of uncertain origin. The ground sense seems to be "that which is not falsified." Meaning "free from pretense or falsehood" in English is from 1530s.

Etymology Online however actually deems it worthy to list the interpretation of your question as disclaimer on the same page:

There has been a temptation to see the first element as Latin sine "without." But there is no etymological justification for the common story that the word means "without wax" (*sin cerae), which is dismissed out of hand by OED and others, and the stories invented to justify that folk etymology are even less plausible. Watkins has it as originally "of one growth" (i.e. "not hybrid, unmixed"), from PIE *sm-ke-ro-, from *sem- "one" (see same) + root of crescere "to grow" (see crescent). De Vaan finds plausible a source in a lost adjective *caerus "whole, intact," from a PIE root meaning "whole."

(Etymonline)

The folk etymology of the English term "sincere" deriving from "without wax”—sans cire in France, is actually very old and common. During the centuries, etymologists have suggested different fields in which wax was used and to which the term may have been applied, like the one about honey sellers from Ancient Rome who were supposed to shout sine cera to convince buyers that it was pure honey with no wax added.

But as noted here:

These origins have long been challenged by Latin scholars, who say that “sincere” comes from a medieval Latin word, “sincerus,” meaning clean, unadulterated, pure of composition. But - dah! Couldn’t “sincerus” have come from “sine” and “cera”? We shall never know until we get that book I’ve always wanted to see published – the dictionary that gives us the root of the word-root, and for several generations of origin where necessary. (Frank Delaney)

The Grammarphobia has the story:

• This is another of those linguistic legends that make etymologists’ hair stand on end. The word “sincere” has no such origin, but the myth, in one form or another, has been causing bad-hair days for hundreds of years.

The more likely etymology:

• “Sincere,” first recorded in English in the 1530s, is from the Latin word sincerus, meaning “clean, pure, sound, etc.,” according to the Oxford English Dictionary.

• The first syllable of the Latin sincerus does not mean “without.” As the OED says, it may be equivalent to the first syllable in simplus, in which sim means “one.”

The "without wax" usage from sculptures:

• But that old “without wax” myth has lived on—and on and on. We’ve found versions of it dating back to the early 1600s. One of the more recent incarnations comes from Dan Brown’s thriller Digital Fortress (2008):

  • “During the Renaissance, Spanish sculptors who made mistakes while carving expensive marble often patched their flaws with cera—‘wax.’ A statue that had no flaws and required no patching was hailed as a ‘sculpture sin cera’ or a ‘sculpture without wax.’ The phrase eventually came to mean anything honest or true. The English word ‘sincere’ evolved from the Spanish sin cera—‘without wax.’

• Though all the stories claim in the end that “sincere” comes from “without wax,” the details vary widely. Sometimes the people trying to disguise flaws in stone were ancient Greek quarrymen, sometimes Roman sculptors, construction workers, or architects.

The "without wax" usage from pottery:

• In at least one version, the flawed goods were pieces of pottery that wouldn’t hold water unless they were secretly repaired with wax. In another, we’re told that a biblical injunction (“Be thou sincere!”) literally means “Be without wax.”

The "without wax" usage from furniture:

• Yet another version, from the early 1900s, claims that “in the days when they began to make furniture,” dishonest cabinet makers used wax to hide the knots and cracks in inferior wood.

The "without wax" usage from writing:

• A gullible writer in 1870 passed this one along: “In old times, people used to write notes to each other, and tie a string around them, and seal the ends of the string with wax. When friends were intimate, and open-hearted toward each other, they folded the letter, and, leaving off the string and wax simply wrote the word ‘sincere.’ ” Hence, he wrote, the Latin for “without wax” became the English word “sincere.”

...and from honey sellers:

• But the oldest versions of the myth claim that vendors of honey in the markets of ancient Rome cried “sine cera” to assure buyers that their honey was pure and free from wax.

Etymological origin from past references:

• Here, for example, is the explanation offered by John Gill in A Complete Body of Doctrinal and Practical Divinity:, Vol. 3 (1796): “The Latin word sincerus, from whence our English word sincere, is composed of sine & cera; and signifies without wax; as pure honey, which is not mixed with any wax.”

• And here’s a definition of “sincere” from a religious dictionary published in 1661: “Sincere is that which is without mixture, as hony without wax.”

• Believe it or not, there are still other versions, but you get the idea. As the OED says, “There is no probability in the old explanation [from] sine cera ‘without wax".