The wait-and-see approach is a common one. On the other hand, some people do need to be operated on before the aneurysm gets to 5.5. It is a serious operation, and the risk of possibly having an adverse surgical outcome has to be weighed against the risk of doing nothing. There are so many variables to be considered in the timing of surgery that if your husband is not trustful of the care he is getting currently, he needs a second opinion from an expert aortic surgeon.
The individual surgeon's skill plays a big role in how safe or risky the surgery will be. Also your husband's state of general health is important in assessing his surgical risk. Since he is in good health now, he might be a good candidate to go ahead and this repaired before he gets any older, and his surgical risk goes up with increased age. On the other hand, maybe he is going to be able live the rest of his life out and die of something totally unrelated, without having to undergo surgery on this aneurysm.
Unfortunately, this issue can't really be settled for you by people who are writing in on a forum. You need to have direct contact with an expert. You are asking good questions, though. If you can get a consultation with a surgeon who specializes in surgery on the thoracic aorta, that would be your best bet. Surgeons specializing in the thoracic aorta tend to be located at major heart centers . You might google "aortic center" and see if there is one near you.
Good luck in getting the answers you need.
Sometimes there is tissue abnormalities in the region of the aorta...if the aorta is bicuspid (bicuspid syndrome)...and the root and valve are affected and often treated at the same time. An aorta root that is greater than 4.0 cm it is considered an aneursym and if it reaches 5.0 cm surgery begins to be an option to prevent a rupture. Also, a higher risk is if the size increases rapidly from 4.0 to 5.0 cm which you report isn't the condition.
The main treatment is to keep blood pressure low and avoid any heavy lifting. I don't believe there is any considerable risk to wait...I have read sizes much higher that didn't present a problem, but you should follow the advice of a doctor who has a more detailed medical history for your huspand. .