JavaScript (ES6), 22 19 23 22 bytes
a=>!/[^2,]|22/.test(a)
Test it
f=
a=>!/[^2,]|22/.test(a)
console.log(" "+f([2])+": "+JSON.stringify([2]))
console.log(" "+f([2,2])+": "+JSON.stringify([2,2]))
console.log(" "+f([[2],[2,2],2])+": "+JSON.stringify([[2],[2,2],2]))
console.log(" "+f([])+": "+JSON.stringify([]))
console.log(" "+f([[],[]])+": "+JSON.stringify([[],[]]))
console.log(f([1])+": "+JSON.stringify([1]))
console.log(f([22])+": "+JSON.stringify([22]))
console.log(f([2,2,2,1])+": "+JSON.stringify([2,2,2,1]))
console.log(f([[1,2],2])+": "+JSON.stringify([[1,2],2]))