“Any skills you’ve previously trained which fall under the Alpha set can be used at any time, regardless of Clone State”
If you let you account lapse into the Alpha state, you can use any skill that is in the Alpha set of skills which are listed in the devblog you linked. That means you cannot use a mining barge, even if you trained for it as an Omega, because the skill ‘Mining Barge’ is not in the Alpha set of skills. Other skills, like say ‘Amarr Battleship’ are now in the Alpha set of skills and you can indeed use them.
Definitely not all of the skills you can train or have trained as an Omega clone can be used while an alpha clone. There has to be some advantage for going Omega.