My mother has sleep apnea and falls during the day. Her oxygenation drops at night to 60%. Why would they avoid adding oxygen to her cpap & daytime?

Depends on reason. Does her oxygen drop to 60% when she stops breathing, or does it drop to 60% and stay there even when she is breathing normally during sleep? If the desaturation is because of sleep apnea, adding oxygen would not help as air is not going to the lungs. Opening the airway with CPAP allows normal breathing and oxygen values. If it is sustained desaturation, oxygen is usually added.
Constant high CO2. One of the reasons that a physician might avoid giving her o2 is that it may stop her breathing. Our main way to control breathing is through sensing of co2. If a person has had a constantly high level of co2 over times due to any respiratory disease, then their body sees that level as normal. Sudden drop in that co2 level can lead to not breathing at all. Please consult your pcp for more info.
Possible COPD. Patients that have COPD may do worse with oxygen at too high a dose.
Does she need it? If her oxygen level on CPAP is normal she does not need it at night. If her daytime oxygen level is normal, she does not need it during the day. Ask her doctor to check the oxygen level during exercise or after a 6 minute walk test. She might actually need the oxygen. An overnight oxygen saturation monitor might be helpful.