Welcome to the forum! No, you are not being irresponsible, they are lying to you. Plain and simple. For one thing, at four months of age, a baby tiel has already fledged so the 'the cage needs to be small' is bullcrap UNLESS the bird is severely clipped in which case you need a cage that is not high at all (because without wings, the bird can fall and hurt itself if the distance to the floor is big) - but I do hope that you are not even considering getting a clipped bird!
Now, the cage that you consider big is not really that big - not for a cockatiel. They need flight cages (meaning both horizontal AND vertical room) so although the depth and height of this cage are OK, the width is not because a bird needs to be able to flap its wings at the very least five times from one end to the other of the cage and, at only 24 in, the bird would not. The other thing is that these high but narrow cages are always too close to the floor and tiels do not like this. I recommend this kind of cage for a tiel:
https://www.amazon.com/Products-Wrought ... way&sr=8-8Also, please get two of them. Cockatiels are intensely flock oriented birds and, in all honesty, they are not happy unless they live with several other tiels but, a companion or mate goes a long way toward satisfying their needs. A lone cockatiel is not a happy bird and an unhappy bird is a stressed out bird and a stressed out bird is an unhealthy bird. So, please, get another one so your bird can have a reasonably happy and healthy life.