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For example, I am confident that very few students majoring in pure mathematics can write a complete proof to Abel–Ruffini theorem (there is no algebraic solution to the general polynomial equations of degree five or higher with arbitrary coefficients) by the time of their graduation. I suspect many students with Master's degree or Doctorate in pure mathematics could not prove this theorem as well. They may know the conclusion, but may not be able to sketch an idea of the proof, let alone give a complete proof.

My question is, should we educate pure mathematics major students in such a way that they should know how to most of the classical results in mathematics such as Abel–Ruffini theorem and Fundamental Theorem of Algebra before getting their bachelor's degree, or at least their master's Degrees?

For the project, you would need to define what are "classical results in mathematics".
I suspect that different people would disagree on the classicalness of various results.

Furthermore, it is doubtful if everyone should learn the same classical results - mathematics is a big field. You can, of course, cut it down by a sufficiently strong definition of "pure mathematics", in which case, maybe everyone should know the same results.

I think it would be more fruitful to ask what results a particular university or study programme should consider as key results, concepts and tools. This nicely sidesteps the logistical issue of coordinating mathematics study programmes worldwide and the potentially ugly definitional issue of who qualifies as a pure mathematician. Furthermore, since it retains the current situation of different mathematicians knowing different tools, it widens the scope of overall knowledge. Collaboration with and research periods at foreign universities, or at least universities with different priorities, is useful partially due to such differences.

There is the further issue of how much one can realistically ask of bachelor (or even master) students. As Steve Gubkin mentioned in a comment, many bachelor students have problems with even simple proofs. Depending on how wide one wishes to cast the net of "classical results", it might be unfeasible to teach them to everyone, without making "everyone" a much smaller set then nowadays.

This is an interesting question, but, understandably, confounds at least two different things. E.g., is it really the case that to "know" a true mathematical fact is to be able to produce its proof on command? I think not. Another diagnostic question: must we understand thermodynamics and the Carnot cycle to drive a car usefully? Must we be able to prove the stability of the proton before setting our coffee cup on the table? Yes, of course, I'm exaggerating... but my exaggeration is in the direction I think is relevant.

Namely, awareness is the key point (and assimilation of the facts into one's world-view... to the extent that they might have some impact and affect one's own decisions).

My opinion on this is in the same vein as my objection to people being told to do every exercise before moving forward: not only are many of those exercises either make-work or pranks, but many are also incomprehensible without understanding what happens in the sequel... which one will not see until after? A bit perverse. Sure, some such pranks are "fun" in Math Olympiads and Putnam and such, but...

The problem that I see is that undergrads are too often conditioned to be paranoid that there's some unfathomable flaw in what they've written... that can only be adjudicated by the oracular professor. One of the worst corollaries of this is that kids are very inhibited about broadening their scope, because they're already worried about defending themselves with regard to a tiny, trivial "plot of land", and are taught to give no credence to their own critical faculties.

So, yes, I think this question raises some good issues, but is literally a bit mis-aimed in its assumptions.