TLDR: Your municipal water supply is safe and clean; to ensure that it's likely also full of chlorine, which these filters, marketed with heavy FUD, are often designed to remove.
"The GE website suggests that a water filter is a good idea to avoid exposure to unfiltered water and sediment, inadvertently offering a sad commentary on public water infrastructure and government funding priorities."
No, they're very much advertently creating FUD about water quality just like everyone else who sells retail water filters in developed countries. If, like more than 90% of people in the US (I'd assume it's even higher in the UK), you receive your water from a municipal supplier, that water is tested several times a year and the results are available to you from your supplier (some agencies mail out reports to every customer). You will find that the overwhelming majority of you receive water that is clean and safe to drink.
By contrast, if you have a well or are one of a very small number of people whose municipal agencies are known not to supply clean water, you almost certainly have invested in a properly-sized point-of-entry filtration system; a tiny point-of-use filter is an inefficient and inadequate solution to that problem anyway, and may not be designed to remove the specific type of contamination you have. If you're one of a much larger number of people who have older or poorly-installed plumbing in their own houses, you may well have sediment, heavy metals, and other contaminants in your water, but (a) it's nothing to do with the government unless you live in public housing and (b) you may not even be aware of it because unlike the municipal supply there is no requirement in most jurisdictions that your water be tested at point of use (by your landlord or the person who sold you the property). Depending upon the nature of the problem, these little point-of-use filters may or may not correct the issue, but of course they do so only for that single spigot. If you actually have a significant sediment problem, your pipes and fittings will be filled with it and subject to erosion and premature failure regardless of whether you filter it at point of use.
The actual reason you want a carbon-block or granulated-carbon filter for drinking water at point of use is not to remove sediment or make the water safe, though it does indirectly reflect those "government funding priorities" you bemoan: it's to remove the chlorine most municipal agencies add to guarantee the safety of your water. While the levels are safe to drink, chlorine doesn't smell or taste very good, and most people prefer water that has not been treated with it or has had it filtered out. Carbon filters do that quite well, and that's why most of the little point-of-use filters you see retailers offering are carbon filters. They will remove some sediment as well, but that's usually not their primary function. There are point-of-use filters designed only to remove sediment, but these are typically only the very cheapest models and are of little use to most people.