It's really sort of unclear how exactly water stays radioactive for 200 years anyway. I mean, the emitters that are suspended in the water are much heavier than water so you can isolate them just through evaporation. Literally all you would need to get drinking water in Capital Wasteland is "run a dehumidifier".
Plus, the GECK works by turning kinds of matter into other kinds of matter, and the purifier includes a GECK, so I don't know why you wouldn't just turn the radioactive waste suspended in the water into, say, inert carbon or something. Maybe James and Dr. Li just weren't very good at their jobs.
Could you explain the former point a bit more?
And for the latter point- how do you know that's the GECK isn't modulating the water?
Okay, so basically radiation is "something unstable emits a particle" (there are three kinds of ionizing radiation depending on which particle is released) and the way this damages your body is that it's a high energy particle on a microscopic scale, and that's going to do damage at a cellular level. There's only two ways for water to be radioactive: the water itself is radioactive (i.e. the hydrogen in your H2O is Tritium) or it contains heavier unstable particles (Carbon-14, say) so that ingesting or being around the water subjects you to radiation from that source, not from the water itself.
It's entirely implausible that the water in the Potomac basin is mostly tritium, since tritium has a half-life of like 12 years, and it's only produced in the upper atmosphere in small amounts from cosmic ray bombardment or from working reactors (you make it by bombarding Lithium-6 with neutrons in a working fission reactor.) So there's not going to be anything that conceivably produces a significant amount of tritium and dumps it in the river.
So the only way the water is plausibly radioactive is "it contains unstable isotopes of things other than H2O". So there's say, Carbon-14 and Radium-226 in your water supply, so drinking it and being around it is a bad idea. But here's the simple solution: Water boils at 100 degrees celsius, Radium boils at around 1100 degrees, whereas carbon boils at around 4800 degrees. So if you heat water to its boiling point you're going to get nothing but pure H2O in the steam, which you can collect and condense to get pure clean water. Since evaporation on the atomic level basically works through individual molecules being excited enough to "go ballistic" and boil, pretty much all of the water vapor in the atmosphere is going to be clean too, so you could just run a standard dehumidifier that you could buy at any department store before the war.
Alternatively, you could always just filter water through dirt too, since the clay in the dirt will bind with all the radioactive particles and what comes out will be purified (this is what the American Civil Defense Organization recommends in the case of nuclear war: get a 5-gallon bucket, punch holes in the bottom, put 2" of pebbles on it, cover with a porous cloth, put 8" of clay on top, cover with a pourous cloth, top with another 2 inches of pebbles. Then pour water through it letting it drain into a clean container, replace the clay every 10 gallons or so.) The fact that radioactive particles will get trapped in the clay makes "purify the land so we can grow stuff to eat" a much more interesting plot than "purify the water."
But the problem here is that Fallout 3 treats radiation like it's magic. In a fantasy game, like the Elder Scrolls, we can only understand how magic works based on the information given in the text, because magic is not a real thing, so when something doesn't make sense we can wave our hands and say "that's just how the magic works, I guess." But radiation is a real thing, and people who play the game (like me) may know a thing or two about how radiation works, so when the game doesn't bother to explain how this works (which it doesn't, except through the mechanics for gaining/losing Rads, which are an okay if gamey abstraction), we can actually drill down on this by saying "wait a minute, I have a degree in physics..."
As for the latter point, the GECK is designed to transmute blasted irradiated landscape into a verdant clean garden. Fallout 3 itself states "The G.E.C.K. will collapse all matter within its given radius and recombine it..." So it's a hyperscience qua magic device that turns one kind of matter into another kind of matter. So if you're running water with unstable isotopes in it through the GECK, why aren't you turning them into something inert? That's what we're told the GECK does- it transmutes matter. So there's no reason there should be a large amount of radiation present waiting to be released in the Jefferson memorial, as the actual radioactive particles will just lose their energy if contained and become harmless helium, electrons, or light, and the emitters theoretically held in suspension in the water could be processed by the GECK.
The long and short of this is that you shouldn't do plots about "radiation" in Fallout games unless you're prepared to show a bunch of leaking hazardous waste casks or some other plausible source for that radiation. It's been 200 years, so all but the 7 long-lived fission byproducts from the bombs should have been rendered almost completely inert (and these are a very small portion of a given fission reaction, like .1%) You can justify the presence of the nasty 7 if you show vats of glowing stuff, but not in the general environment just handwaving "there was a nuclear war 7 generations of ago." If you want to invoke technology or a scientific phenomenon as the cause of something, go ahead and posit something that doesn't actually exist (a fission battery, a way to preserve brains in gel, FEV, etc.) Don't go invoking something that actually exists, make it work some way it can't actually work, and don't bother explaining how it's supposed to work in this context.