I didn't see any pile of rust, did you mean that slight amount of combustion debris in the picture of the gas burners?
There is no question that you could improve the efficiency of the system by getting a modern boiler but from what little I can see on the pictures it appears the existing boiler has been well cared for so I think I would get another opinion before going the route of a new boiler based on a little bit of combustion debris that looks entirely normal.
BTW, the installer did NOT do a Manual J if all he did was measure the lengths of the baseboards and counted the number of windows. Gathering the information for a complete Manual J would normally take about an hour and includes the area of all the external wall, area of the floor over unheated spaces, the area of the ceiling below unheated spaces, area of windows AND the total linear feet of the perimeter of all windows and doors along with several other items. The installed heat emitters (baseboards and radiators) is not used in a Manual J but could be a factor in determining if a modulating/condensing boiler would be appropriate.
Without being there and making my own inspection I couldn't state if the existing boiler might last ten years or ten months. I will state that even if the boiler is in like-new condition you could save an awful lot of fuel over ten years by going to a modern boiler.
You need several more evaluations and estimates for replacement before making a decision.