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 of verses.
My point is, it obviously was never a big deal or important matter. The relativeness of time was always intended, since, if God wanted precise timing he would command his messenger to create a clock for his followers! so when he didn't, he must have never intended precise timing!
Saying so is simply wrong (even if there's some truth in your words, but as written they are strictly speaking wrong and misleading)!
Allah intended us to pray at five specific times a day
You can't say that Allah never intended a precise timing. First note that if we were asked to pray at a specific time none of us would be able to do it this would be an extreme hardship, so it would go against many verse of the qur'an saying that our religion is ease not hardship!
But Allah indeed intended a well defined timing as the prayers are related to specific and well observable and distinguishable times (or more correctly speaking time periods) of the day. (I guess that all of these times are quoted in the answers of Does the idea of praying 5 times a day come from the Qur'an?).
And as these timings are well defined and known they are precise (enough) for the believer. As he knows he should seek refuge in the prayer at five specific times a day in which he can communicate with his Lord and Creator.
Time is important in our religion you can't perform Hajj all the year, you can only fast Ramadan when the new crescent moon has been sighted and you can only pray a prayer when its time has started. And you should have finished doing so before the time for the next prayer starts. Doing it right on time is rewarded as a good deed doing it wrong might be sinful.
All these times are usually observable
And no Allah never said something like "there was no means to know what time it was", but he indeed gave us a hint how we could make out fajr saying:
… And eat and drink until the white thread of dawn becomes distinct to you from the black thread [of night]. Then complete the fast until the sunset…. (2:187)
In fact many ahadith report that the sahabah did use white and black threads for the given purpose. There certainly may be cases when these prayer times can't be observed, but in case of doubt Islam offers again some ease, if we have done efforts and means to find out the time for the prayers we have fulfilled our obligation (Alhamdulillah).
There's no need for clocks in Islam, but Muslims may need them
Time is a relative term. So clocks are not necessary if you don't need 100% precision and that is the case with the prayer times. Nevertheless we are humans and in our faith we always want to do everything in the best and most accurate manner, even if this knowledge would lead (some of) us to delay prayers or even join a lot of prayers at the end of the day. But we want to have rules and limitations to get the feeling of worshiping Allah in the best manner and that's why we need specific timings: to have something we rely on or we could assign the blame to.
One could assume that the differences in the calculation methods which are spread all around a world expresses the leniency of the time periods, but I'm persuaded that the issue is rather the fact that they are more accurate if used in the right location. As there doesn't seem to be a universal formula. (See also Figuring out right calculation method of prayer time)
So certainly the definition of these timings clearly say there's a time span and the limits are somewhat lenient: the transition from the end time of one prayer to the beginning time of another may take 5-10 minutes or in case of the transition from fajr (fard) or sobh to dhohr some hours. one could say the inaccuracy of the calculated prayer timings and the leniency of the timing limits defined by the known day times create a certain balance.
Why knowing the time is important
Nevertheless it is essential for Muslims to know these limits and it is essential for the (Muslim) ruler to have well-versed people placed in mosques who would call for the prayer at the very beginning of the time. So in some countries some of these people rather use their eyes observing the sky than clocks to call for it, while others rely on time tables. As Allah the almighty says:
… Indeed, prayer has been decreed upon the believers a decree of specified times. (4:103)
One could say that the call for the prayer -the adhan- is crucial as hearing it means one must pray the prayer which was called for and one must have finished the prayer which was before. As 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)
So a prayer is only valid or more exactly performed on time if at least the above criteria applies (and this is a time gap of very few minutes). And other rulings can be concluded from that for example if a woman's menses ended in this small time gap described above the actual prayer is a due on her else not (see for example Check if menstruation stop before fajr) also fasting starts and ends with the adhan. So if one goes on eating after the adhan has started a fast is invalid etc.
In Islam the prayer times are well-known and distinguishable time periods of the day. And each prayer must be performed within its given time Periode. This is the ease Allah gave us, we may have at least around 1 hour to perform our duty which usually wouldn't take more than a few minutes. A 100% precise timing is not needed, but as many of us layman don't know these timings or even many more rely on the adhan it is essential for a Muslim ruler to have people call for the prayers at the very beginning of the time, even if we should know that the precision of such a time lays somewhere between +/-1 minute and +/-5minutes.
We should also have in mind that worshiping Allah is a due and even if Allah gave us some clear rules (time, direction, purity, how to etc.) it is for our own best to perform it and we shouldn't look for excuses to delay them.