Ignoring polysynthetic and agglutinative languages (which I have no wish to do other than for the purpose of the question I pose) languages may be placed somewhere on the synthesis/ analysis continuum. Languages such as Sanskrit, Greek (Ancient) and Latin would come very near the synthetic end of the continuum and languages such as Chinese and Thai would come very near the analytic end of the continuum. English is rather nearer to the analytic end than the synthetic end. I believe that there is a theory that languages move from synthesis to analysis; this can clearly be seen in the way Latin changed into French, for example.

I have read many of the threads in this forum with interest. What has struck me is how quite often two well informed people cannot agree on how to analyse a language question and how many questions seem to be incapable of satisfactory resolution.

My question therefore is this: Is it the case that the more a language moves along the continuum to analysis, the more elusive it becomes and the more difficult to analyse?
I really don't know anything about it. But off the top of my head, it would seem the more independent words a language has, the more complex it may be. Instead of having a more limited set of base words, the analytic language has a large sets of words which evolve in their own ways, independent of any base. If satisfactory mutates, unsatisfactory mutates with it. But if non-X is independent of X, the two words may, over time, no longer be (true) opposites. Similar arguments would apply to all prefixes, suffixes, and other word modifications.
what do you mean by analytical language and kindly site some examples?
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<My question therefore is this: Is it the case that the more a language moves along the continuum to analysis, the more elusive it becomes and the more difficult to analyse?>

I don't know, but would you imagine that all theorists agree on how to analyse a language question that is related to Latin usage?
I would define an analytic language as one in which the functions of words in a sentence are defined either by other "grammatical" words (and not by the changes of endings in words) and/or by word order.

Languages are not necessarily either analytic or synthetic, but exist on a continuum.

One of the features of analytic languages is that meaning can sometimes only be established by context. Here is a real life example:

I have a kit of items for cleaning compact discs, vinyl records and cassette players. One item is described as a "compact disc cleaner" (for cleaning compact discs) and another as a "carbon fibre cleaner" (for cleaning vinyl records). The two constructions appear to be similar. Without context we know that "compact disc cleaner" must be something to clean compact discs because we can discount the possibilty that it is a cleaner made of compact discs. However, we need the context to know that "carbon fibre cleaner" is a cleaner made of carbon fibres and not something intended to clean carbon fibres. If someone comes onto this forum asking what "grammatical" rules there are to explain what "carbon fibre cleaner" means we are hard put to find them. Other languages may make a clear distinction between "a cleaner (made ) of carbon fibres" and "a cleaner for (cleaning) carbon fibres.
I do not. Perhaps due to the bias I have as a native English speaker, I believe concepts that rely on the establishment of the relationships between people and objects, objects and other objects, and people and time are most easily understood universally through one or (as is often the case in German-English dialogue) more than one preposition, rather than through affixes. In the large, meanings are understood better when the item that distinguishes its meaning is largest, as usually prepositions constitute larger morphemic units than do affixes, which essentially seek to accomplish the same goal: establishing relationships between people, things, and time (among many other aspects of grammar). Take, for example, German. In German, "I gave her to you yesterday" would be "Gestern gab ich sie dir", when there is no "to" to indicate the indirect object, making it easy to confuse whether the speaker gave you to her, or her to you. So, in short, my answer is that analytical languages are more easily understood by a speaker of any language by virtue of its very nature, that is, that these relationships, whose meanings hinge on the function of affix or preposition, manifest themselves in the form of a preposition rather than an affix.

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I am not sure your German example is a good one. The indirect object can be expressed in English without a preposition: Give the man a medal. Give me it etc.

More generally, I do not think it is the case at all that languages that use prepositions rather than case endings are more readily universally understood. It is, as you half suggest, a question of what you are used to. There are some languages that make do with far fewer prepositions than English and they not make use of case endings.

Anyway, the point I am trying to get at is that when the function of a word in an utterance is not indicated either by a preposition or some bound morpheme, the meaning can often only be determined by context or you simply have to know.
In Enlgish we mark case with prepositions and word order and even when the dative isn't marked with "to" or "for" we can always distnguish between the two objects because the dative comes first:

I'm buying some food for them -----> I'm buying them some food

Although I know there are some occasions when we rely completely on common sense, on the whole marking cases with word order and prepositions doesn't take anything away from the langauge nor does it make it harder to analyse. Why does French need countless conjugations for person when the pronouns are put in? English doesn't have them and meaning can be conveyed just as easily.