What could be an alternative meaning to the word "planets?"

Solution 1:

To summarize what he is saying:

We have been given a paper by a convention of men who were sent to draft it. People are now doubting the ability of the same people they sent to write their proposal.

They were charged with making a proposal on how to shape the government that would then be brought to a general assembly for consideration as to its merit. And, if it has such merit, it will be voted upon and adopted.

The convention was given wide latitude to propose anything they felt fit. So, stop questioning whether or not they did what they were supposed to do. It was only their job to propose an idea, not to enact it into law.

Instead, let's vote upon this as if it fell from the skies and wasn't written by people you may or may not have respect for. Let's just consider it on its face value alone.

He's saying in effect, shut up and vote. And, more importantly, vote as if you don't know who wrote this, but only rely upon the words that are written. Pretend it dropped from the planets, fell from the skies, appeared out of nowhere, etc. Just judge it on its own merits, not those of the convention that wrote it.

The second half of your question:

The people have the right to form the government. Their representatives are not necessary for that process.

Mode here means model. The mode of government is the model of government. Out of context, he appears to favor a true democracy over a republic. But, this may just be rhetoric.

Solution 2:

The image, that of something dropping from a planet, used by Pendleton is perhaps better understood in the context of Ptolemaic, geocentric astronomy in which the Earth was fixed and everything else revolved around it. In that context planets were wandering stars, referred to by Greek astronomers as ἀστέρες πλανῆται or stellae errantes. The word was not confined to the planets of our solar system. In such a model, the Sun or the Moon were also considered as planets.

In Pendelton's time, Ptolemaic astronomy had long been replaced by the Copernician heliocentric model. But it is probable that, in the late 18th century, the image of something strange falling from the sky from a moving celestial body, a planet, passing above us was still easily and readily understood.