Once Texas declares and wins its independence, is the U. S. free to annex
it? Why would Mexico object to the American annexation of Texas?

i believe george gallup used a poll in 1936

i believe it would influence their own way of understanding things. it shall be a foundation where they can draw or start an argument.   historians are also  scientists that  have their tools of analysis and expertise to unlock a variety of difficulties such as relationships of man, origin, politics, economy, and culture. based on facts and understanding on these areas, they may shade light to extract ideas and develop options to validate a hypothesis or debunk it.