Your question is referred to in the review Beis Hillel כסלו תשס''ה under the title “Why does not a kohen make a blessing שעשני כהן".
Rav Asher Anshil Schwarz asked the question and offered several answers. Relevant to your question is:
From the Mogein Shaul
He also cannot make the blessing “שלא עשני ישראל ” because a kohen is
included in the designation ישראל. Further, he would be glorifying
himself in the shame of his friend because until the sin of the golden
calf, the temple service would have been in the hands of the