How to explain the need/advantage for volatile in terms of microcontroller instruction flow
Short answer: no, volatile cannot be explained in terms of the
figure you show in your question.

Longer answer: there is no direct link between the meaning of volatile
and the architecture depicted in the figure. If you insist in linking
them together, it is possible to find a very weak and indirect link
as follows:


Volatile is all about preventing the compiler from doing some
optimizations when translating your program to assembly. Looking at the
generated assembly can help understanding the effect of volatile.
When learning the assembly language, the figure you show is helpful
in understanding the assembly instructions themselves.


As you see, the link is indirect, mostly because we are talking about
different abstraction levels. Also, the link is weak because the figure
describes specifically the AVR architecture, whereas volatile applies
to any architectures for which there is a C compiler.

To illustrate the first point above, here is an example program which
shows the effect of volatile:

volatile bool volatile_flag;
bool plain_flag;

void setup()
{
    Serial.begin(9600);
}

void loop()
{
    Serial.print("Waiting for volatile flag... ");
    while (!volatile_flag) ; /* wait */
    Serial.println("done.");

    Serial.print("Waiting for plain flag... ");
    while (!plain_flag) ; /* wait */
    Serial.println("done.");
}


As such, the program is not very interesting, it will just hang on the
first loop. But you can imagine there being some ISR that can set the
flags when some external event happens. Then the wait loops would start
to make sense, as they wait for the external event.

Here is the disassembly of the loop() function:

    Serial.print("Waiting for volatile flag... ");
  ce:   60 e0           ldi r22, 0x00   ; 0
  d0:   71 e0           ldi r23, 0x01   ; 1
  d2:   82 e5           ldi r24, 0x52   ; 82
  d4:   91 e0           ldi r25, 0x01   ; 1
  d6:   0e 94 b3 02     call    0x566   ; 0x566 &lt;Print::print(char const*)&gt;
    while (!volatile_flag) ; /* wait */
  da:   80 91 51 01     lds r24, 0x0151
  de:   88 23           and r24, r24
  e0:   e1 f3           breq    .-8         ; 0xda &lt;loop+0xc&gt;
    Serial.println("done.");
  e2:   6e e1           ldi r22, 0x1E   ; 30
  e4:   71 e0           ldi r23, 0x01   ; 1
  e6:   82 e5           ldi r24, 0x52   ; 82
  e8:   91 e0           ldi r25, 0x01   ; 1
  ea:   0e 94 d0 02     call    0x5a0   ; 0x5a0 &lt;Print::println(char const*)&gt;
    Serial.print("Waiting for plain flag... ");
  ee:   64 e2           ldi r22, 0x24   ; 36
  f0:   71 e0           ldi r23, 0x01   ; 1
  f2:   82 e5           ldi r24, 0x52   ; 82
  f4:   91 e0           ldi r25, 0x01   ; 1
  f6:   0e 94 b3 02     call    0x566   ; 0x566 &lt;Print::print(char const*)&gt;
    while (!plain_flag) ; /* wait */
  fa:   80 91 50 01     lds r24, 0x0150
  fe:   81 11           cpse    r24, r1
 100:   01 c0           rjmp    .+2         ; 0x104 &lt;loop+0x36&gt;
 102:   ff cf           rjmp    .-2         ; 0x102 &lt;loop+0x34&gt;
    Serial.println("done.");
 104:   6e e1           ldi r22, 0x1E   ; 30
 106:   71 e0           ldi r23, 0x01   ; 1
 108:   82 e5           ldi r24, 0x52   ; 82
 10a:   91 e0           ldi r25, 0x01   ; 1
 10c:   0c 94 d0 02     jmp 0x5a0   ; 0x5a0 &lt;Print::println(char const*)&gt;


The important thing to notice is how the compiler implemented the
waiting loops. The first one is essentially

0:  lds r24, volatile_flag  ; load the flag from RAM
    tst r24                 ; test it
    breq 0b                 ; if zero, loop back


whereas the second loop is:

    lds r24, plain_flag     ; load the flag from RAM
    cpse r24, r1            ; if zero, skip the following jump
    rjmp 1f                 ; jump over the next loop
0:  rjmp 0b                 ; infinite loop
1:


You may appreciate the first loop has been translated to assembly in a
very literal way, whereas in the second loop the compiler optimized away
all but the first read of the variable. That could be translated back to
C as

if (!plain_flag) {
    for (;;) ;  // hang in an infinite loop
}


The compiler assumed the flag cannot be changed by some external event,
so there is no point in reading it repeatedly. The volatile keyword
prevents the compiler from making such assumption.

And how is all this related to your figure you may ask? Well, if you
happened to not understand the assembly code above, you could check the
AVR Instruction Set Manual to see what each instruction does,
and that's when the figure becomes helpful.