(1) I have not found anyone who discusses why the custom is not to do kidushin . with a shtar. However I assume the reason must be a practical one - it is simpler to do kidushin with money. If one does kidushin with a shtar, one has to make sure that one gets everyone's name correct (with a get, which of course must be done on a shtar, there are often problems concerning how to spell the names etc.) so the custom grew up to do kidushin with a ring. One sees from the Rosh Kidushin 1:8, who writes that the custom of the world is to do kidushin with a ring that does not contain a precious stone, that it is an ancient practice to mekadesh with a ring.

(2) If one did perform kidushin with a shtar, the kidushin would be valid even though it goes against the custom. This is because, mideoraisa, kidushin with a shtar is valid so a mere custom does not have the power to uproot the Din of the Torah.

However it should be pointed out that there are many halachos concerning how to write a shtar kidushin - there is a whole chapter in Shulchan Arukh (Even HaEzer #32) about this - so if one got it wrong the kidushin might be invalid, or more likely there would be a doubt whether it is valid and if the lady wished to marry someone else she would require a Get.

(3) There is a contemporary ramification of shtar kidushin - namely, civil marriage. I found in Teshuvas Minchas Yitzchak 1:12:3 that he cites the Levushei Mordechai who writes that the secular marriage contract might have the validity of a shtar kidushin, so, again, a woman who had a civil marriage might require a Get if she wished to remarry.

Dovid Bloom

Here is a better answer to your question why the custom is not to mekadesh with a shtar:

1. The answer is that the present-day custom seems to be based on Kabalah.

The Rema in Shulchan Arukh Even Ha-Ezer 27:1 writes that the custom is to mekadesh with a ring, and there is a reason for this in Tikunei Zohar.

As a result of the custom to use a ring, the use of a shtar was therefore abandoned as popular custom.

2. Another reason - this time according to the revealed Torah rather than the hidden Torah - is given by Sefer HaChinuch, Mitzvah 552. He writes that part of the thinking behind the Mitzvah of getting married is that the wife should always remember that she is connected only with her husband and be faithful to him. This way the husband and wife will live together peacefully.

For this reason the custom is to perform kidushin with a ring, because this is always on her hand to remind her of her husband.

In contrast, if kidushin is done with a shtar, the wife does not have this constant reminder.

Kol Tuv

Dovid Bloom