Why do they go to the minister to be married to begin with? There are a variety of reasons.
1. They are divorced and choose not to go through the annulment process. Thay have many reasons for that, from delay, cost, fear of refusal, feeling that to declare their earlier marriage invalid is somehow deceptive, etc. Many believe their first marriages were valid when they were performed, but various circumstances that arose later caused that marriage to disintegrate. Searching for a flaw in that first marriage can make them and/or their children very uncomfortable, almost as though those children were not quite legitimate.
2. Many Protestant ministers - not all - are not seriously troubled by the divorce issue in certain situations. Since they themselves are married, they are likely to understand the struggles marriage can bring. They usually inquire as to why the couple were divorced and then decide whether or not to perform the ceremony. Many are ready, say, to remarry someone who was married first when very young, or the partner strayed, or abuse of some sort, etc.
3. Many Protestant ministers - not all - do not require attendance ar a pre-Cana event. They may require (often) three or four conferences with the pastor instead.
4. Often, and especially if the marriage is a mixed one - say, Catholic groom and Protestant bride - the Protestant (and maybe the Catholic, too) is alienated by certain rules governing Catholic marriage - e. g., the Catholic promising to exert religious influence on his partner, to try to raise the children Catholic, rules re who can be witnesses, etc. Maybe the Protestant is the one more involved in his/her church.
5. Millions of Catholics are of the cafeteria variety, and when they find a likely mate who is committed to some other religion they have no problem switching. (Mainline Protestants often don't speak of Catholics as 'converting' since they are likely to be ecumenical and not hostile toward other Christian groups. Evangelicals emphasize being 'saved' and developing a 'personal relationship' with Christ, and conversion means that rather than changing church affiliation.) Protestants seldom herald converted Catholics as 'Journey Home' heralds those who convert to Catholicism.
6. Many ministers interpret the command of Christ differently. To begin with, he did not say that divorce and rermarriage is an unforgiveable sin. In addition, they regard the teaching of Christ in its context as a defense of women. At that time, as I recall, Jewish men could say to their wives three times "I divorce you" and that was it. They see Christ's words as mainly a response to this severely unjust, patriarchal tradition.
7. Let's face it. An annulment usually is seen as a church-sanctioned divorce. It is widely regarded as an escape valve by which someone can remarry with the blessing of the church. I've read that they are much more likely to be granted in some dioceses than in others, which can raise additional questions.
8. What about prolific adulterers like JFK - or that convert from Georgia who hopes to be president? It can seem hypocritical that because G's first two marriages were not performed by a priest his divorces are acceptable, glossed over. There is something very offensive about that - at least in my mind. Is it somehow less of a sin when an adulterer is married by a minister than by a priest?
9. Some ministers who marry divorced people think this way: regardless, it's better to marry this couple and keep the church's foot in their door than to have them go to a JP - and possibly carry with them some resentment against the church generally. This means a better chance that they will have a church wedding and stay in the church once married.
God bless everybody - no exceptions. And a blessed Christmas season to one and all!
I know people in here who have been Catholic for years and are now becoming Protestants and they’re some of brightest, most loving, godly people I know. 
. It was like a one-way ticket to hell, thanks to the powerful catechetical efforts of the nuns when we were young. After all, these people weren’t converting to anything, they had their whole world fall apart in a most gut-wrenching and soul-killing way, and that on top of the financial mess and the shame (in those days, there seemed to be more embarassment over failed marriages than today).
), etc. etc. As GKC says, history and theology are complicated!