You can do that but the drawback of that design would be wasting power for the 5V rail (heat dissipation for the linear regulator). 12 V AC is going to be around 15 V DC. So the 5V regulator would have to dissipate (15 V - 5 V) * 1.2 A = 12 W. And you need a regulator type rated for more than 1.2 A. It would make more sense to use a buck converter for the 5V rail.
Yes, I know that there will be quite a bit of power waste but the computer I am building this for is not getting much use so it's not really a problem. The heat dissipation issue is more of a problem but can be dealt with using a large enough heatsink.
But aside from the power waste is there any other concern with this kind of solution?
The 7805 5V regulator was just an example, I will probably use an LM323K instead which is rated for 3 Amps which is a bit overkill but it looks nice and has a lot of capacity. Some 7805's can be used for up to 1.5 Amps so they should work too if properly cooled.
The good thing with this solution is the simplicity, smaller PCB footprint, less number of components used etc.
The reason I don't want to use a switch mode PSU for the 5V rail is because I want clean power with as little ripple noise as possible to avoid any potential damage to the old IC's used.