Salamun Alaikum,
The Quran simply gives us 5 'periods' of the day to pray. These prayer periods are 'prescribed' (4:103) and as believers we would be expected to get the best meaning from the Quran (39:18, 39:55) and simply do the best we can (64:16). Man indeed has long fought over insignificant details which have nothing but choked believers.
Our responsibilities are to join or to make an appropriate congregation for prayer if we can. With regards the Shafi and Hanafi position with regards Asr, I would 'discern' between the two positions from the Quran.
I read from the Quran, the requirement for Asr as a late prayer which can be read up till sunset. This is clear in my humble opinion from the Arabic, 'wa-qabla gharub' (before the setting)
050:039
“Therefore be patient of what they say, and sing the praise of your Lord before the rising of the sun and before the setting (Arabic: wa-qabla gharub)”
From the above verse, we note the reference before rising (Arabic - qabla tulu-i - i.e. Fajr) and before the setting (Arabic - Wa qabla l-gharubi). The prayer before the setting of the sun in the late afternoon is Asr.
Also, we note in 20:130 that the two ends of the day (watarafa l-nahari) are used as separate periods along with the period just before the setting of the sun (Shamsi waqabla ghurubiha) which clearly indicates the period of Asr prayers.
020.130
“Therefore be patient with what they say, and celebrate (constantly) the praises of thy Lord, before the rising of the sun (Arabic: Qabla taluhe- Shams - i.e. Fajr), and before its setting (Arabic: Qabla gharubiha - i.e. Asr); yea, celebrate them for part of the hours of the night, and at the sides of the day (Arabic: Watarafa l-nahari - i.e. Fajr and Isha): that you may have (spiritual) joy”
Therefore, I would understand 'Asr' as a late prayer (before sunset) so I would personally be inclined to join the 'Hanafi' traditional prayer which for me is the later out of the two options available (Shafi and Hanafi).
I am also of the opinion that one prayer ends when the 'congregation' for the next prayer begins.
I also completely concur with your statement:
"However, I have a feeling that this also may be unnecessary, as it is possible to determine the period of prayer just from looking outside and having an estimate of the period of the day".
I am sure that is what many of the Bedouin Arabs did to whom the Quran was initially revealed.
In the end, it is comforting to note what requirement is expected by God (Do the best you can). On the other hand, what is required by the mullah clergy is another matter.
064:016
“So keep your duty to God as best you can / what you are able (Arabic: ma is’tata’tum), and listen, and obey, and spend; that is better for your souls. And whoso is saved from his own greed, such are the successful”
I hope that helps, God willing
Joseph.