Hot water heating recommendation
This veers towards being "a matter of opinion" but you would seem to be of a mindset where a "heat pump waterheater" and perhaps "yet another minisplit for the family room" would be most suited to your direction of progress.

An oil boiler which is not heating much of anything but your hot water has a lot of "standby losses" - i.e. keeping itself warm while doing nothing. The heat exchanger coil that heats your hot water may be suffering from mineral deposits, and if so could be improved by cleaning or replacement, but if that's pretty much all your boiler is doing, it's a sledgehammer being used to drive tacks.

Various other alternatives are options, and the choice of "best" really comes down to your opinions/preferences - change to an oil-fired water heater (probably with tank - very few "tankless" units seem to exist in the market) and perhaps use a heat exchanger off that to provide the limited heat when needed to the family room - a system sized far closer to your actual needs than an old boiler that used to heat the whole house.

If you happen to have natural gas available at your house, it's generally a better cost than oil per unit of heat, and tankless water heaters are commonly available. Of course, if you do, it's nutty to have an oil rather than a gas boiler, but I've seen that sort of nutty...