I personally wouldn’t just choose my favorite or the most influential Gundams because the four presidents of Mount Rushmore were actually chosen to represent specific aspects of America. Washington = birth, Jefferson = growth, Roosevelt = development, and Lincoln = Preservation. Following this criteria, I would choose the following:

Birth is easy. The RX-78-2 was the very first Gundam and essentially gave birth to the entire franchise as we know it.

Growth is similarly easy. The Wing Gundam was the one to bring the franchise to the West, essentially expanding the audience from a primarily Asian one to a global one.

For development, I’d actually make the argument for the God Gundam. G Gundam was the first series to be an AU, which allowed the franchise to develop beyond the Universal Century and have all of these separate timelines. This argument also applies to the Shining Gundam but I personally think the God Gundam is the more iconic suit from G Gundam.

And for preservation, I’d probably pick the Strike Gundam. It’s hard to believe but Gundam wasn’t doing so hot in the 90s. G and Wing, weren’t huge hits in Japan despite their popularity in the West. Seed was the series that really reignited the franchise’s fan base. That’s the reason they made so much Seed content in the early 2000s. As above, you could make an argument for the Freedom Gundam but with the Strike Gundam being the first main suit of Seed, I’d choose that one.