Hi,
It’s best to take the eggs away from her and discard them. They are not going to be viable anyway, since they have been moved between a towel and cage grate. Each time they are moved to the grate, this lets them cool off. Eggs can sometimes survive cooling once, but not more than that. A towel is not a good thing for her because they do not like a soft surface like that, and her nails could get caught up in the threads. Also, your concerns are very valid. Chicks that hatch on a cage floor or anywhere but a nest box tend to end up dying. The hen can’t keep them together, or warm enough. She needs a wooden nest box that is made for cockatiels – one can be ordered online. You attach this to the outside of the cage, as high as possible. Or it can go in the cage. Both parents will spend time in the box. They rarely use any nesting material. Do not try to make a nest box or any type of nest – these will not work and you will end up with dead chicks. The box has to be very large, enclosed, wood with the right size entry hole, in the right place. Once she lays eggs, one parent sits during the day and the other sits at night. You have not mentioned a male, so he must not be helping. This is another reason the eggs are not going to hatch. Unless you have no male, in which case these eggs are not fertile.
Thank you for asking Lafeber,
Brenda