Why can't the Pope have children?

The truth is that there were Popes who had children.

Pope Pius II (1458–1464) had at least two illegitimate children (one in Strasbourg and another one in Scotland), born before he entered the clergy.[10] Pope Innocent VIII (1484–1492) had at least two illegitimate children, born before he entered the clergy.[11] According to the 1911 Encyclopædia Britannica, he "openly practised nepotism in favour of his children".[12] Girolamo Savonarola chastised him for his worldly ambitions.[13] The title Padre della patria (Father of the Fatherland) was suggested for him, precisely with suggestions that he may have fathered as many as 16 illegitimate children.[14] Pope Clement VII (1523–1534) had one illegitimate son before he took holy orders. Academic sources identify him with Alessandro de' Medici, Duke of Florence.[15][16] Pope Gregory XIII (1572–1585) had an illegitimate son before he took holy orders.[17] [edit]Sexually active after receiving Holy Orders Pope Julius II (1503–1513) had at least one illegitimate daughter, Felice della Rovere (born in 1483, twenty years before his election). Some sources indicate that he had two additional illegitimate daughters, who died in their childhood.[18] Furthermore, some (possibly libellous) reports of his time accused him of sodomy. According to the schismatic Council of Pisa in 1511, he was a "sodomite covered with shameful ulcers."[19]

Pope Paul III (1534–1549) held off ordination[20] in order to continue his promiscuous lifestyle, fathering four illegitimate children (three sons and one daughter) by his mistress Silvia Ruffini. He broke his relations with her ca. 1513. There is no evidence of sexual activity during his papacy.[21] He made his illegitimate son Pier Luigi Farnese the first Duke of Parma.[22][23] Pope Pius IV (1559–1565) had three illegitimate children before his election to the papacy.[24]

Pope Sergius III (904–911) was supposedly the father of Pope John XI by Marozia, according to Liutprand of Cremona in his Antapodosis,[25] as well as the Liber Pontificalis.[26] However it must be noted that this is disputed by another early source, the annalist Flodoard (c. 894-966), John XI was brother of Alberic II, the latter being the offspring of Marozia and her husband Alberic I. Hence John too may have been the son of Marozia and Alberic I. Bertrand Fauvarque underlines that the contemporary sources backing up this parenthood are dubious, Liutprand being "prone to exaggeration" while other mentions of this fatherhood appear in satires written by supporters of late Pope Formosus.[27]

Now, to be fair, many of these were in 1400's, a time that I suspect even most Catholics would now call the low point of the papacy, especially in theological terms. There were three Popes at one time, all sorts of politicking, and in general, the Papacy was just another earthly Kingdom. (Again, I want to keep stressing even modern Catholics would probably agree with me on this.)

Additionally, the first Pope, Peter, was at least known to be married. (Jesus healed his mother in law, after all).

Celibacy is not a "modern" innovation, but it is clearly post-Biblical.

It's very simple actually. The Pope can't have children because in Western Catholicism, Priests are expected to be unmarried and celibate.

Mary and Joseph had a child despite celibacy, but they were at least married (thus giving Christ a legitimate father). I can't see how someone who cannot have an illegitimate child (celibate) or a miraculous child (not married) would be able to accomplish this, but I suppose that anything is possible through Christ.