feeling the pulse

What about feeling the pulse? Is it crucial for interpretations of classical music? 
It is crucial. If you don't have the pulse in every cell of your performing body, your playing will just be controlled by single (and therefore overcharged) parts of your muscular apparatus which leads to a cramped sound. As a consequence this sound leads to cramped listeners. So, If your whole body can refer to a pulsing orientation line in your mental picture (your "inner ear"),  strained movements of your playing can be flexibly unbent.

But it is not a kind of an inner metronome. the pulse is flexible itself and determined by the will of your interpretation (or, unfortunately' by the will of a conductor too ;).
And no..., this does not mean that any rubato time or strong arbitrary shifts in tempo are always desirable (although I think, more courage to work with rubato sometimes could be fine). The nub of it all is a relaxed inner pulse-feel making up all of your phrasing, your attacks, your accents - just every musical parameter of instrumental or vocal technics. It is your "timing" which is bound  to the pulse. 

And this  inner feel of pulse is the very presence of making music itself, no matter what style you play. There is no delightful phrase of a melody, no rousing rhythm structure, no compelling musical communication without a vivid feel of pulse between performing musicians. 
Again this does definitively not mean, that music has to be regular or bound to certain rhythmical structures to convince its listeners. We can listen to periodical sounds in nature which do not refer to steady pulse as waves or birds or other animals communicating in an unperiodically recurrent way. Yet you can become aware of a mysterious pulsing in these nature sounds: there is a 'timing' and a kind of feel for this timing. 
To outline my thoughts in other words: timing and pulse is a magic sensation, both for musicians and for listeners. It makes us simply feel alive, really alive and not forced into culturally prescribed rules or rhythms.