Does unifying the times of the Adhan and Iqama to achieve congregational gathering fulfill the meaning of the Hadith: "The most beloved deeds to Allah are prayers performed on time"? And is praying Asr at 3:45 PM still within its time?

It is sunnah to call the adhan at the beginning of the prayer time. It is permissible to delay it so that it is closer to the time of the iqamah (call to commence prayer), but it is not permissible to call the adhan before the prayer time begins, except for Fajr prayer, because the adhan was legislated to announce the beginning of the time. If there are muezzins in a town, it is preferable for them to call the adhan at the beginning of the time to avoid creating chaos and disagreement.

The optional time for Asr prayer ends when the sun turns yellow, and delaying it beyond that is not permissible except in cases of necessity. Asr prayer at 3:45 PM is before the sun turns yellow.

As for Dhuhr prayer, it is not permissible to call the adhan for it before its time begins, as the Umm al-Qura calendar shows that its entry time changes throughout the year. If the adhan is at 11:45 AM, it might be before the time begins on some days, which is impermissible by the consensus of scholars.

The preference for praying at the beginning of its time has exceptions: cooling down (ibrad) with Dhuhr prayer in intense heat, and delaying Isha prayer if it does not cause hardship. Dhuhr prayer, according to the Umm al-Qura calendar, will be at the beginning of its time throughout the year. As for Asr prayer on some short days, it will be after more than half of its time has passed, but it is permissible as long as it is before the sun turns yellow.