I really think your main issue is that the male is too young to breed, as I have already mentioned. And it sounds like the female may be too young. The male needs to be at least one and a half years old, but two years is better to be sure he is emotionally mature enough for breeding. Your female needs to definitely be two years old – not “maybe 1-2 years old”. If you encourage a female to breed when she is too young, she is at high risk for becoming egg bound and dying. Young birds do not make good parents and are more likely to develop bad habits like breaking the eggs, eating the eggs, or just not sitting on them consistently. And if they do hatch chicks, they are more likely not to feed the chicks or to kill the chicks, either by accident or on purpose. Once these bad habits begin, the birds have been ruined as far as being good breeders.
I think there is also the possibility that you are not correct on their sexes. You said both were lutinos and the female had spots under her wings. I would check both birds again and make sure the male does not have any spots and the female does. Cockatiels can keep female markings for as long as a year. If they are not on a good diet, they may not have an adult molt at the right time, and they may still appear to be female.
Breeding birds is not something to rush into. Most importantly, it takes a LOT of patience on your end and you will be disappointed in the beginning. It is rare for first time parents to successfully raise chicks. It can take several attempts over a year or two before they have success, if they have success. First they have to bond, then they have to effectively breed or the eggs will be infertile. Both adults must be fertile and interested in breeding. Then they have to incubate the eggs properly. Some pairs start to fight at this point. If they do manage to hatch any chicks, they might not figure out how to feed them. In this case you need to be prepared to hand feed the chicks every 2 hours around the clock for a couple of weeks, and you will still have night feedings for at least 2 – 3 more weeks. Then if you do manage to end up with weaned chicks, they cannot live with the parents or with each other. The parents will either start attacking them or when they get older, they will breed with them. Siblings will breed with each other. It is up to you to keep any related birds from breeding. You will need to find homes for them or possibly a pet shop will trade you some food or supplies for them, but they rarely will buy them, and if so, not for very much money.
For now, I think you should split these birds up and not even think of buying another bird. You need to learn a lot more about breeding before you begin this venture. Let the birds live side by side in separate cages until they are older – at least 6-7 months. By then, the male may start showing some interest in the female. You need to make sure they are on a good diet – seed is not an adequate diet for pet or breeder birds. They need to eat a nutritionally balanced diet like pellets or our foraging diets. Cockatiels do well on our foraging diets because they are 100% nutritionally balanced like a pellet but still contain whole ingredients instead of being ground up. They also need to learn to eat dark leafy greens, veggies, fruits and multi grain bread. Once they are set up to breed, you need to add cooked egg to their diet. Breeding birds need a lot of privacy so once they are set up, they need to be in a room where they do not see people very often. If you check on them too much they can become nervous and not want to breed or they may abandon their eggs or chicks if they have any.
I’ll provide a link on feeding your birds a good diet. Then you need to Google how to breed cockatiels and research as much as you can. You need to know everything to expect before you set up a pair of birds to breed. But keep in mind, you can do everything right on your end and it still may not result in breeding or chicks being raised. This is where patience really comes in. If you think this is going to be easy and everything will just happen as it should, then you probably need to choose a different hobby and just have birds as pets. This is not to discourage you, but breeding birds is a lot of work and can also be expensive. It is not something to do on a whim or without a lot of research.
Bird Food Guide
Good luck and we are here whenever you have a question,