Only three football rules in use today remain unchanged from those laid down by the Intercollegiate Football Association (IFA) in 1876. Every other rule was created or evolved to address specific problems or otherwise improve the game. In many cases, the concerns that led to creating a rule are no longer relevant, but the rule proved useful for other reasons and remains in effect today. Such is the case with the longstanding penalty for roughing the punter. The rule was not put in place to protect the punter during the act of punting, but to protect the punter after he punted the ball. Why would football need to protect the punter after kicking the ball? To make that connection, we must step back to understand punting and the concept of "onside" during the early days in football's history.
You got the general idea right, but nearly all the details (especially the years) wrong. Let's see how many I can straighten out:
Until the changes of 1900, you didn't have to be onside at the time of the kick, only at the time of the recovery. That is, the kicker could put you onside AFTER kicking the ball by going past you upfield. The change you state as having been made in 1880 was from 1900, which was also the date of the roughing-the-fullback rule's introduction, for almost the reason you gave. Also beginning in 1900, the kicker, although onside, was ineligible to recover the ball. The three changes were to go together, removing the incentive and adding the penalty.
7 on the offensive line were not required until 1906. In 1903, a minimum of 6.
The other form of onside kick you describe both came in and went out later than the dates you gave, and required the ball to hit the ground beyond, and later 20 yards beyond, the line. The original form of onside kick didn't require the ball to hit the ground for recovery, and it continued until 1923, while the later version was abolished a few years later; they coexisted.