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Compared to selecting a proper cast iron radiator and having an experienced steam pipe install it, Electric wires are easy! Operating cost will be somewhat higher. And since you are already paying to heat the steam boiler, that difference can be substantial in the colder months. But this is only one room and therefore affordable. Just don't get the bright idea to do the hole house and scrap the steamer…

And I remember back when my parents were raising me, if I happened to touch a hot radiator and get a burn, out parents would say, "I guess you wont do that again"

So I would consider the parenting style of today and recommend the one that is high on the wall. By the time the toddler is tall enough to reach that hot radiator, hopefully they have learned to keep their fingers away from hot things.

Since it is an "up stairs" room, I would assume that the floor has a warm room below it, and the radiant value of the high on the wall radiator will work perfectly in your case. As far as how one works in comparison to the other, The rules of physics don't change from one radiator to another. One kWh of electricity will make 3412 BTUs. Can't change that.

You have a hand fired coal boiler that has been converted to burn natural gas.

You have huge amount of thermal mass in the boiler and radiators you will never have that in a new boiler unless you add storage on the return header entering the boiler sump.

Have the gas burner serviced and cleaned for now and leave the system as is.

That boiler looks great. And there are very few moving parts that make it operate. Mainly the limit control has something that expands and contracts as the temperature changes, As does the original thermostat (if you still have that) and the gas valve has a coil that is an electro-magnet the opens the valve when everything is cold and closes the valve when things heat up. That valve coil makes the main burner gas turn on and off.

Any new equipment will have electronic timers for ignition and combustion, circulator pump to move the water that needs no pump in your existing boiler, a combustion fan, and all sorts of limit and pressure switches to monitor when the gas should burn and when the gas should stop.

By the look of that Honeywell Gas Valve, you have one of the safest standing pilot controls available. That valve is replaceable with standard off the shelf parts you can source locally of on amazon.com.

So your homework is to take the cost of the 2 or 3 free estimates for a replacement boiler and compare it to about 30% of your annual gas usage. (the gas usage for winter only, not the laundry, cooking or hot water). To get that number for heat only use your lowest gas bill and multiply by 12. Call that number X. then take a years worth of gas bills and figure out that total. Call that number Y. Now deduct X from Y and that will be your heating gas usage bill. Multiply that number by 0.30 and call that S1. That is what you will save with a new heater in one year. Multiply S1 times 10 and get a total of 10 years of savings. (That is savings not adjusted for inflation). Compare your 10 year savings to the cost of the new boiler. If the boiler is a lower cost, then replace the boiler. If the boiler is a higher cost, then keep the old boiler.

I selected 10 years because most new boilers will last between 17 and 30 years with an average of about 20+ years. So that leaves you at least 10 additional years after you have recovered your initial investment to actually get a return on your investment.

Unless you only plan of being in your home for a few years…. that will skew the whole thing

Edit: The savings of 30% are conservative. I know from experience that you will save more than that. I have done that job with both Atmospheric cast iron boilers and with the new modulating condensing boiler. The savings will be more like 40% with the Cast Iron, and closer to 50% with the ModCon. The problem is that the job is so expensive between asbestos abatement, scrapping the old iron from the site, and building the proper near boiler piping for the replacement, It ain't cheep! If you are into DIY, then you are better off with a new boiler. If you are going to pay retail, then you need to seriously look at the numbers.

Very doable and worthwhile!

Keep up the TLC and that will be going long after any of us are around — let alone the modern equipment.