Is a Real-Time Clock (RTC) necessary for real-time systems?

The article you linked is just complete and utter nonsense. The "real time" in "real time clock" (as it's used to refer to the type of hardward device described in the article) and the "real time" in "real time systems" are completely different terms. The former means storing the current calendar time (usually some very poor approximation of it, as opposed to high-precision like the linked article claimed) and advancing it without external power, using a long-life button/coin type battery. The latter means responding to events with hard bounds on latency from the time of the event to the time of the response.

A few other bits from the article, to establish that it should be regarded as untrustworthy:

Almost negligible. Of the order of 1 sec in 100 years

1 sec in 100 years is roughly 317 ppt (yes, that's parts per trillion). You can't get that kind of clock stability with any existing commercially-available clock technology. Even getting it to 1 second per year would require at least an OCXO which requires a high-power, always-on oven regulating the temperature. The idea you could get it with a device powered by long-life coin battery is laughable.

real time systems like digital clock, attendance system, digital camera

None of these are what one would call real time systems.

Real Time Systems are something that responds to an internal or external event /stimuli in a specified time and that time is usually in milli or micro seconds. It needs timer of small precision rather than RTC.

And the answer to your question is No, it won't affect the real timeness of the system.

If your system is offline after reset and having RTC, it will be able to put proper dates into the logs. Logs could be huge in case you need to look through them and having wrong timestamp will make you, your software developers and clients crazy, and in general investigation almost impossible.

Easy or difficult, low or high in the article you refer to is a kind of personal opinion. It is difficult and costly if you never did it before and do not have clear system requirements and statement of work; and it is easy and cheap when you know what you need and what is the best device to be used.