You pretty much hit the nail on the head here. If a couple is Halachically married, and they don't get a Halachic separation, they remain married and all further relations are adulterous (with all the negative results that come with it).
This is a straightforward Mishnah (first one in Kiddushin) and no later authorities disagree. The Mishnah says (stripped down to the relevant parts):
הָאִשָּׁה נִקְנֵית בְּשָׁלשׁ דְּרָכִים, וְקוֹנָה אֶת עַצְמָהּ
בִּשְׁתֵּי דְרָכִים. נִקְנֵית בְּכֶסֶף, בִּשְׁטָר, וּבְבִיאָה. [...]
וְקוֹנָה אֶת עַצְמָהּ בְּגֵט וּבְמִיתַת הַבָּעַל.
A woman is acquired [for marriage] in three ways, and she acquires herself in two
ways. She is acquired through money, through a document, or through
sexual intercourse. [...] And she acquires herself through a bill of
divorce or through the death of the husband.
The Mishnah is pretty straightforward: if a woman is Halachically married, the only way she could Hlachically un-marry is by a Get or becoming a widow.
Some have argued that if a woman never Halachically gets married, then she doesn't need to get unmarried through a Get. Indeed, I believe it was R' Moshe Feinstein who argued this opinion. When Jewish members of the Soviet Union started arriving not having had any formal Halachic marriage/divorce, he ruled that there was no issue of Mamzerim with those marriages since there was never a Halachic marriage to begin with.
As a result, there are opinions that argue if one is attending/officiating a wedding of an irreligious couple, there may be a long-term benefit in ensuring that the marriage is not done in a Halachically correct manner, to prevent any future possible occurance of Mamzerim.
However, not all opinions agree with that approach. If you look back at the Mishnah, you'll see that the third manner in which a woman can be married is through marital relations. Some authorities (among them R' Yosef Eliyahu Henkin and R' Joseph Ber Soloveitchik, I believe) felt that the fact that a couple was living together as husband and wife was enough of a proof that they intend to be married as a husband and wife, and therefore are considered Halachically married.
According to this opinion, even if the couple never married in a 'formal' Halachic ceremony, they are still Halachically married, and as a result can only divorce through a Get. Therefore, there is no benefit to not having a Halachically correct ceremony, so if one is attending/officiating at such a wedding, might as well make the ceremony as 'correct' as possible. Any problems that may occur in the future would be a problem anyways, so there is no benefit to not having a proper wedding now.