Why is X(jw) real and even if x(t) is real and even
But he seems to have ignored the differential substitution, because when making the variable switch, since dτ=−dt The whole term on the left would become negative, and we would find the opposite of what the author intended to prove,


You missed that he also reversed the limits of integration.

The initial limits were from \$t=-\infty\$ to \$t=\infty\$. When substituting you would first end up with an integral from \$\tau=\infty\$ to \$\tau=-\infty\$. Reversing the order of the limits to get back to an integral from \$-\infty\$ to \$+\infty\$ cancels the negative sign on the differential element.