Old house, new boiler, new flue, new connection to the old chimney. All good.
Except for a couple of things.
First, the hole beneath where the flues connect to the chimney has been filled with mortar.
That is not good.
Second, the new connection to the chimney required that the old connection from the water heater inclines downward. That impedes exhaust drafting.
That is not good, either.
On top of the hip roof, the chimney is on the side of the house. It is very high, too high for most ladders. And dangerous to access from the roof side. It is covered with a flue cap, which is good. But if something inside the flue, like bricks or spalling ceramic flue liner, should become dislodged and fall, how could they ever be removed? That clean-out port was there for a reason. To fill it with mortar is short sighted, at best. It is dangerous at worst.
If enough material should fall, it could conceivably clog the flue and the boiler/water heater's ability to discharge exhaust gases. The resulting back draft into the house could be deadly. It would be deadly over time - carbon monoxide is unkind.
Chipping that mortar out, putting in a new tube to allow easy access and sealing it tightly with a cute cover (tin plate, acorn, pineapple) would be an easy and good fix! And very Colonial!
My recommendation: When you see new installations, it is sometimes hard to know if they are installed correctly. Time to call a home inspector!