Your argument is popular, but incomplete. Let's look a the First Amendment:
Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances.
The second amendment could easily have been constructed in a similar fashion, I'll write my own "what if":
Congress shall make no law prohibiting the ownership of arms.
Nice, simple, and would support your interpretation. However that's not in the historical record. In fact, the Second Amendment was not only written differently but was passed with different text by congress than the states used to ratify! Here are the two versions:
A well regulated Militia, being necessary to the security of a free State, the right of the people to keep and bear Arms, shall not be infringed.
A well regulated militia being necessary to the security of a free state, the right of the people to keep and bear arms shall not be infringed.
So where the first amendment is an absolute prohibition "no laws", the second amendment uses an arguably gentler "shall not be infringed". To put that in a frame of mind, consider something like having to get a license. Under the first amendment a license to be part of the press would clearly be no good under the "no laws" clause. Under the second amendment, is having to get a license infringement or not? To a lot of the public it is not.
It is also interesting that they saw it necessary to include the concept of a militia. I won't attempt to guess what they really intended there (although plenty of others have), I will just point out the language is a marked departure from the absolute, unabridged nature of the first amendment.
In short, assertion would require that there be striking similarity between the two causes, such that a limit on "arms" would have the same parameters as a limit on "freedom of the press". But the two clauses are not only dissimilar, but completely different. I think based on text alone it is entirely reasonable to make the general statement that "the founders viewed the ownership of arms differently than freedom of the press", otherwise the much simpler text of the first amendment, or indeed adding "arms" to the first amendment, would have been far, far simpler.