There is parts missing or could be just me not being able to play good...

It is just you not being able to hear that the tune is essentially repeated, just with the arrangement getting richer every time. Rawlins Cross do a perfectly professional job, and their arrangement is of course their intellectual property. If you want to play the tune, do it your way.

"Pretty close": such tiny differences are always left to the performers' taste. Here is an ABC of the beginning as played by RC:

X:1
T:Memory Waltz
M:3/4
L:1/8
K:G
Bc | d4 de|d4 c2| B2 Bc dB| G4 D2 | E4 G2 | c2 cd e2 | A2 B2 c2 |

...