This is not a proof of the inequality, but simply an answer to your question.
Indeed, you can replace the $2022$ in the numerator with $ab+bc+ca$ to arrive at an unconstrained inequality that must be true if the original constrained inequality is true. To see this, make the substitution and assume for sake of contradiction that the substituted unconstrained inequality is false for some choice of $a, b, c$. Because the substituted unconstrained inequality is homogeneous, you can scale each $a, b, c$ by an appropriate constant to cause the constraint to become satisfied. But those scaled values then imply that the original constrained inequality is false, a contradiction. Thus, the substitution is justified.
After performing the substitution, the inequality is indeed homogeneous, and so the variables can be scaled arbitrarily without changing the value of the LHS.