If you ever saw a 60s Godzilla movie you might notice the trick effects are ridiculously bad even for back then. Godzilla for the Japanese is a personification of natural disasters like earthquakes and tsunamis which hit Japan frequently, but also of the atomic bombs .
For the Japanese sense of humor those 60s Godzilla films are hilariously comic and they will laugh constantly when watching one. It takes the fear and worry out of those events and helps them to get over with them.
That is not exactly true.
The Showa Goji (the 60s Godzilla movies as you called them, but through '74) has actually originally a horror movie, very serious, very clearly about the atomic bomb (the american release cut some of those scenes and added some others), and technically sophisticated for the time. In fact, the first few movies were fairly serious. Later, somewhat related to the release and popularity of Gamera from another studio, Godzilla became more heroic and in some movie a bit goofy. But it would be a miss characterization to say that the character was comic relief even in the later movies. Also, in the later part of the Showa era series, godzilla pretty much lost any connection to being a representation of atomic destruction or a natural disaster.
Then, Heisei Goji ('84 - '95) Godzilla was brought back again generally in a more serious villain role, though she lost most of her connection to atomic disaster and was more a "force of nature." The Heisei series pretty much maintained its serious tone and godzilla as a villain (with humans coming up with new ways to kill it), though she did help humanity out a couple of times by defeating other monsters.
Millennium Goji ('99 - '04) is a bit all over, but it was definitely not mad for comic relief. GMK ('04) is a serious film with Godzilla being the incarnation of malice of the souls the Japanese killed during WW2.
Finally, the Reiwa Goji ('16 - ) is a serious movie mostly about governmental dysfunction (with the prime minister and his cabinet being destroyed even) and the notion that Godzilla was spawned from environmental / toxix waste (not unlike one of his enemies, Hedorah, in the Showa era).
So, to suggest that Godzilla was simply comic relief is not true. That was more an American sentiment than a Japanese one, though not completely absent from the character at times. Additionally, what Godzilla "represents" has changed many times depending on the needs of the director, studio, or just to reflect the times.
Myself, I've watched Godzilla since the '70s, been to Toho studios, and mostly watch the movies in Japanese. I may be a bit defensive about the Big G, but I can assure you that your characterization is a vast over simplification.
I do not think you could stat out a natural disaster, and even the Tarrasque cannot compete.
I think you can give Godzilla stats, in fact I've done it in 4e and 5e. It is just outside the normal limits provide by WotC.
EDIT: Also, to suggest the special effects were bad for the era is not true. Suit-mation was originally chosen because the makers of the movie thought the technique was
superior to stop motion animation, being more fluid in movement with a great sense of mass. I do agree the quality dipped as the studio pumped out more and more movies. The even started using stock footage to save cost in some Showa era movies.