Are Mr. Mattson's recommendations the only way to write a literary analysis essay?
Nope.
Should my thesis always be based on the meaning of the work (the theme statement)?
Must my supportive points always be different types of support? Like irony, archetypes, symbolism, etc.?
Should my thesis always be based on the meaning of the work (the theme statement)?
Mr. Mattson believes this is one of the best ways, but it's not the only way. You should though talk about the meaning of the work sometimes in your essay.
You could base your thesis on an interpretive problem or controversey in the text.
Examples
Big Brother plants Julia in Winston's life in order to use his feelings for Julia to torture him and to make Winston into what Big Brother wants.
To fully understand Shakespeare's Hamlet, one must understand that Hamlet has strong romantic feelings for Ophelia, his own mother, and for Horatio.
Base your thesis on something more like the focus of the text.
Hamlet is not so much about the issues of life or death or action vs inaction. Instead, it is about social classes and how these artificial divisions make life more complicated and tragic than it should be. {note the addition of a theme statement}
Orwell's 1984 depicts not just a dystopian society. The novel depicts a hell. Winston is literally in his own personal hell from the start. Read this essay for an example.
Jane Austen's Pride and Prejudice is an example of proto-feminism, but Austen herself can't fully shake her own internalized misogyny.
Golding's Lord of the Flies may be an allegory for the ID, EGO, and SUPEREGO, but, more importantly, it is a comment on toxic masculinity.
Mr Mattson's method of having three types of support is good advice but not always and there are other ways.
SP1
Hamlet - dramatic irony, archetypes and subversions, juxtapositions
SP2
Ophelia - dramatic irony, archetypes and subversions, juxtapositions
SP3
Horatio - dramatic irony, archetypes and subversions, juxtapositions
SP1
What most people think 1984 is about and why that is incomplete.
SP2
Evidence that the Inner Party is demonicly supernatural
SP3
What Orwell's point would be in doing this
SP1
Research about Jane Austen's beliefs beyond P & P
SP3
Evidence from P & P that Austen harbors misogyny
SP2
Acknowledgment that Austen is a proto-feminist in the novel and elsewhere.
SP4
How this idea changes one's interpretation of the novel