I don't think the Enter Sandman comparison is as ridiculous as some make it out to be. I don't know if I'd cite a specific song, but there's a definite Metallica meets DT thing happening in As I Am. Right down to the drum beat.
The drum beat is literally the most basic alternating bass/snare drum beat in existence in rock/metal. I could also point to 100 AC/DC songs with the same drum beat.
It just happens to be a similar tempo to Enter Sandman that also uses the same drum beat.
Musically, it's mostly not Metallica at all imo, aside from the riff. The main riff certainly could be, but OTOH no more specifically Metallica than many other metal bands. Chugging metal riff in standard metal scale (don't know the name of it, but basically minor with the flattened 2nd) with flattened fifth note. That could be any metal band. Yeah, Enter Sandman has a prominent flattened fifth in the riff too, but that's where the comparisons end with that song.
That opening slow riff is straight from Black Sabbath (the song, and the band), and the main riff is a sped up variant of that, inspired by thrash in general, including Metallica who have been one of DT's bigger influences, so comparing that to Metallica isn't ridiculous at all. Enter Sandman isn't a Metallica song I'd cite as similar in style though.
Beyond that, the vocal melodies are nothing like Metallica. Hetfield's signature is grunting out the same few notes (basically the root, third and seventh below the root), while As I Am covers a much larger full octave range covering minor and mixolydian scales for a very different feel to what Metallica does.
And then there's the low-key prechorus with the stick hit drum beat, the softly spoken vocals and the 3 part harmony section, and the half time feel chorus. None of it is reminiscent of Metallica.
There are plenty of DT sections and songs that show strong influence from Metallica, but I've never heard any strong similarity between As I Am and Enter Sandman, or for the most part Metallica in general. Change "As I Am" to "Constant Motion" and get back to me.