No, amssymb is not redundant with the other packages in your MWE. It sets up a number of other math fonts and symbols. Because its names for those fonts and symbols have become so standard, many other packages either load amssymb automatically, or provide their own version of the same symbols.
In contrast, amsmath sets up a large number of typesetting command, but does not change the fonts, amsthm sets up commands for theorems, and amsxtra sets up some rarely-used commands for backward compatibility. Finally, mathtools extends amsmath with additional commands.
If you do not need amssymb because you are loading different math fonts, you do probably want to take it out. The AMS fonts use up several of the sixteen math alphabets legacy 8-bit TeX allows you. Trying to load more math alphabets than that could cause your document to fail, on some TeX engines.
amssymbprovides additional symbols that aren't "named" inamsfonts; it is not redundant. However, since you are loadingmathtools, that loadsamsmath, and you have madeamsmathredundant. – barbara beeton May 02 '22 at 02:25amssymbloadsamsfonts, but neither is loaded byamsmath.mathtoolsloadsamsmathand provides additions to that package. All ofamsmath,amssymbandamsfontsare "protected" against being loaded twice; I haven't checkedmathtools, but it probably is too. Still, it's not recommended to load any package twice. – barbara beeton May 02 '22 at 02:55amsmathbeforeamsthm(ormathtools). See this question. – Sandy G May 02 '22 at 03:31\qedhereis not provided byamsmathnormathtoolsit's defined inamsthmwhich needs to be loaded separately. Note that classes likeamsartloads it automatically – daleif Aug 13 '22 at 14:56