I am not Syrian, nor do I claim to know about why the Syrians do what they do, but this seems to be similar to what was done in the days of Kings David and Solomon.
Yevamos 24a:
ת''ר אין מקבלין גרים לימות המשיח כיוצא בו לא קבלו גרים לא בימי דוד ולא
בימי שלמה א''ר אליעזר מאי קרא {ישעיה נד-טו} הן גור יגור אפס מאותי מי
גר אתך עליך יפול אבל אידך לא
It was taught in a braysa: we do not accept converts in the times of
the Messiah. Similarly, they did not accepte converts, not in the days
of (King) David, nor in the days of (King) Solomon. Rabbi Eliezer
says, what verse teaches us this? (Yeshayahu 54:15) "הֵן גּוֹר
יָגוּר אֶפֶס מֵאוֹתִי מִי גָר אִתָּךְ עָלַיִךְ יִפּוֹל," "Behold, the
one with whom I am not, shall fear, whoever mobilizes against you
shall defect to you;" but no one else.
(translation of gemara is mine; the translation of the verse in Yeshayahu is Judaica Press, copied from Chabad.org)
Similarly, on Yevamos 76a:
ויתחתן שלמה את בת פרעה מלך מצרים גייורי גיירה {מלכים א ג-א} והא לא
קבלו גרים לא בימי דוד ולא בימי שלמה מידי הוא טעמא אלא לשולחן מלכים
"And (King) Solomon married the daughter of Pharaoh, king of Egypt"
(Melachim I 3:1). She must have converted. But how could that be?
Isn't it true that they didn't accept converts, not in the days of
(King) David, nor in the days of (King) Solomon? The answer is that
they didn't accept converts in those times because of "the table of
kings."
Rashi, there (sv מידי):
האי דלא קבלו גרים אלא משום דלא הוו מגיירי אלא לשולחן
מלכים הא בת פרעה לאו צריכא להכי שהרי בת מלך היתה
That that they didn't accept converts was only because those that came
to convert came for the "kings' table" (i.e., to partake of the wealth
of the Jewish people). Pharaoh's daughter clearly didn't convert for
this reason, because she was the daughter of a king.
(above translations mine)
We see from this that we should only take converts at a time when things aren't going so great for the Jews, because then we know that the conversion was for good reason. However, when things are going really well (as in the times of King Solomon, and the Messiah), then converts should not be accepted, because at those times they're probably trying to convert due to "שלחן מלכים," and just trying to take advantage of the Jews' good position in society.
Again, I don't know if this is why the Syrian community has taken up this practice; it's just that the circumstances seem similar to me.
