Allah (ﷻ) specifically pointed out that prayer has specified, fixed times (see verse below). That is where the great focus and emphasis on exact times comes from. Whether you use a clock to keep track of time or the sun, you're still trying to get the exact times and pray on time. Prayer outside of its time frame is rejected.
Qur'an 4:103 فإذا قضيتم ...
The narration you've mentioned shows that among the scholars of Medina covering the mouth was considered as a bad act which shouldn't be done while praying. As Salim ibn 'Abdullah ibn 'Umar is one of the so called 7 scholars of Medina from among the tabi'yn and a grand son of 'Umar ibn al-Khattab.
Even if there are further evidences scholars differed ...
Covering the mouth during Salah is considered a makruh action.
وأن يغطي الرجل فاه وهذا إن كان بغير عذر، وغلا فلا يكره
It is also undesirable for a man to cover his mouth while praying unless he has an acceptable excuse for doing so.
— الفقه على المذاهب الأربعة - Islamic Jurisprudence According To The Four Schools
The evidence for this includes:
We find in the sunnah:
He who finds a rak'ah of the prayer, he in fact finds the prayer.
(In Sahih al-Bukhari and Sahih Muslim and elsewhere)
This is the criterion and basis on which we can decide whether or not we have prayed on time. This is also the basis that decides whether or not we have prayed in a congregation as one variations of the hadith ...
some indicated some methods to get a better idea of time, but even with everything you say, looking at shadows, etc, you get a relative time period and nothing remotely close to precise timing with a clock down to minutes, and even seconds.
More over, at nights(and dawn etc), there was no means to know what time it was, as it is also noted in Quran in one ...
Also I have same problem in my hostel room . So I decided to put a bucket( which I do not use for anything but to wash my feet by pouring with a mug ) near to sink or in a corner .
I do wudu in sink except and finally wash my feet in the bucket..
Hope this may help anybody....Assalamu alaikum☺