How can one achieve contentment with Allah's decree and destiny regarding the gender of a newborn, especially given the weakness of the soul and its susceptibility to the words of others, and the unanswered supplication?

Supplication is beneficial in all circumstances: either the supplication is answered promptly, or it is stored for the Hereafter, or evil is averted by it. Shaming someone for having daughters is a legacy of the Age of Ignorance (Jahiliyyah). Bearing sons and daughters is in the hands of Allah, as He says: "To Allah belongs the dominion of the heavens and the earth; He creates what He wills. He gives to whom He wills female [offspring], and He gives to whom He wills male. Or He makes them [both] males and females, and He renders whom He wills barren. Indeed, He is Knowing and Competent." And it may be that what we dislike is good for us, and what we love is evil for us. Wondrous is the affair of the believer, for all of it is good: if he is afflicted with ease, he is grateful, and if he is afflicted with hardship, he is patient. And whoever is afflicted with daughters, they will be a shield for him from the Fire.