Per comments, OP's instructors are probably just incompetent (or perhaps he somehow misunderstood what they meant). In standard English, and has two distinct meanings...
1: Dick liked Jane and she liked him
2: Dick insulted Jane and she slapped him
...where #1 simply uses and to join two "independent" statements - stripped of pronoun substitutions,...
1a: Dick liked Jane
and (another statement, with no specific relationship to the first)
1b: Jane liked Dick
But in #2 the conjunction implies a temporal/causal relationship between the two statements...
2a: Dick insulted Jane
and (subsequently - pragmatically, probably consequently)
2b: Jane slapped Dick
Grammatically/semantically, the and in #2 can be replaced by and then or simply then. Perhaps OP's instructors felt he was overusing the two-word form in contexts where it was pragmatically obvious that the second event was later in time and/or caused by the preceding event. But this is a matter of style, nothing to do with grammaticality.
TL;DR: Even if OP's instructors knew what they were talking about, they've obviously failed to convey the correct information to their student. The only reasons to avoid and then are (1) - if the context implies no [con]sequential relationship, and (2) - because using it excessively can result in a stilted prose style.
EDIT: In fairness to OP's instructors, they might be (mistakenly, imho) attempting to impose notions derived from prescriptive grammar. Consider...
2a: Dick insulted Jane
2c: Then Jane slapped Dick
2d: Jane then slapped Dick
2e: Jane slapped Dick then
...where 2c-2e are all valid sentences/clauses that could follow 2a. It's irrelevant here whether we put a period or a semicolon between 2a and whichever of 2c-2e we choose to use (you wouldn't hear a difference in speech anyway). The point is the fact that then can be moved around like this implies it's an adverb, not a coordinating conjunction. As that link says...
Be careful of the words then and now; neither is a "coordinating conjunction", so what we say about coordinating conjunctions' roles in a sentence and punctuation does not apply to those two words.
I cant say I find the above caveat particularly enlightening, but it may help explain some of the antipathy directed at the usage and then.
That sentence is grammatically incorrect - and doesn't make sense. I can't tell you why "once" is there - because it should not be!
A correct re-wording could be:
Transporting the stone once it was broken was comparatively simple
or
Transporting the stone when it was broken was comparatively simple
or, with a slightly different meaning:
Transporting the stone when it was broken once was comparatively simple
Best Answer
Short answer
Long answer
You are correct in the overlap between once and since, they can both function as adverbs and conjunctions.
Adverb
Note that since is the adverb of the verb "to make", but in the sentence "I made a cake from meat and have been feeling ill since I made the cake from meat" there are a lot of unnecessary duplicate words. This can be shortened to "ever since", "ever since then", "since then", "since that time". However, the word still functions as an adverb of made, linking that time with now. The phrase "I since made a cake" is not used because there is no way to link then and now. You need to say "since the time I made a cake from meat" - "I felt ill".
Conjunction
Preposition
This indicates a spatial, temporal, or other relationship. Since and After can both be used to indicate a temporal location.
Again, since implies a past event that is still continuing and as such is often used with the Past, present and future perfect continuous tenses
Once however, does not really get used as a preposition because another preposition is often inserted to be more specific. For example, the prepositions are in bold:
So to conclude, there are similarities between once and since, both describe time but once is finite and since is continuous.
I always thought of after as a straight forward preposition but looking at the dictionary definition it is indeed quite versatile and can be used as an adverb as well. "After I made the cake I felt ill". In this context it would mean "immediately after".