What was actually said aside, as for your claim based on (Russian, but also other experts) that such an attack is just fanciful, the DTRA considered it less so in a 2001 assessment titled:
High Altitude Nuclear Detonations (HAND) Against Low Earth Orbit Satellites ("HALEOS")
It looked at
Vulnerability of commercial and government-owned, unclassified satellite constellations in low earth orbit (LEO) to the effects of a high-altitude nuclear explosion. [...]
One low-yield (10-20 kt), high-altitude (125-300 km) nuclear explosion could disable -- in weeks to months -- all LEO satellites not specifically hardened to withstand
radiation generated by that explosion.
(Emphasis in original.) There are [a lot of] technical details in the presentation, but although you seem to want discuss those here, I suggest physics SE or space SE for that. Here's a semi-relevant slide though:
As was discussed there and in a few other articles though, Russia may have qualms about about doing something like this, because it could/would affect its own non-hardened satellites in almost equal measure. It's speculated that someone like North Korea, which has little space/orbit presence, is more likely to do something like this, in some circumstances.
Scientific American ran an article on this in 2004. They don't quote any dissenting voices, but one concurring:
Dennis Papadopoulos, a plasma physicist at the University of Maryland who studies the effects of HANEs for the U.S. government, puts it slightly differently: “A 10-kilo-ton nuclear device set off at the right height would lead to the loss of 90 percent of all low-earth-orbit satellites within a month.”
And several others in fact concur that commercial satellites are unlikely do withstand that (albeit no being as commited to the specifics):
Hardening satellites is a costly endeavor, however. Greater
protection means more expense and more massive protective
materials. And heavier satellites cost significantly more to
launch. Just in the design phase, hardening efforts add 2 to 3
percent to the multimillion-dollar price tags of satellites, Defense Department sources affirm. According to some estimates,
installing the shielding panels and hardened components and
launching the extra weight can add from 20 to 50 percent to
the total cost of a satellite. Finally, electronic components capable of withstanding the high radiation levels of a HANE —
about 100 times as great as natural levels — offer functional
bandwidths only about one tenth the size of those offered by
commercially available processors, a fact that can raise operating costs by an order of magnitude. [...]
Spacecraft can be protected in other ways, says Larry Longden of Maxwell Technologies, a company that shields satellites.
Sensors can be installed to detect the presence of harmful radiation. A satellite equipped with such a device can be cued to shut
down its computer processors and electronic circuits to wait for
the destructive episode to pass. Despite the risks to civil orbiters,
however, the Defense Department so far has failed to persuade
U.S. satellite builders to harden their spacecraft voluntarily,
states Barry Watts, senior fellow at the Center for Strategic and
Budgetary Assessments.
There's also a discussion there of a project to facilitate more rapid depletion of the radiation belts in the aftermath of such an attack, which sound a little sci-fi but was demonstrated experimentally.
And if you prefer Chinese research US one, they also claim they can do it, and even avoid creating radiation belts (YMMV on that bit) by detonating a much bigger (megatons) nuke at lower altitudes (80km) and rely on the rising radioactive cloud to catch the satellites:
At the Northwest Institute of Nuclear Technology, a research institute run by the People’s Liberation Army in Xian, researchers developed a model that can evaluate the performance of nuclear anti-satellite weapons at different altitudes and yields with unprecedented detail and accuracy.
The simulation results suggest that a 10-megaton warhead – modestly powerful by today’s standards – could create a serious threat to satellites if it detonates at an altitude of 80km (50 miles).
The blast could turn air molecules into radioactive particles and produce a cloud with a shape similar to an upside down pear, said nuclear physicist Liu Li and his colleagues in a paper published in the peer-reviewed journal Nuclear Techniques on October 15.
In about five minutes, the cloud could rise to an altitude of nearly 500km and spread over an area of more than 140,000 sq km.
“The strong residual radiation of the debris cloud may cause failures of spacecraft moving in it, such as satellites, or even cause direct damage that can lead to destruction,” the researchers said.
A space-based explosion would not produce much of a cloud because of the lack of air. High-energy particles generated by the event would mostly be captured by the Earth’s magnetic field and spread around the globe as a radiation belt, threatening a wide range of spacecraft. This could render nuclear weapons ineffective and too dangerous for an anti-satellite mission.
(Funny how something can be both "too dangerous" and "ineffective".)
But because of the presence of air molecules in the Earth’s atmosphere, an explosion in near space would create a cloud with a total mass far greater than the bomb itself, according to Liu.
“Due to the high concentration of fission products inside the debris cloud, the released gamma rays and beta particles are strong, making their effects on spacecraft and communications within the affected area stronger,” Liu’s team wrote.
Immediately after the blast, the cloud would rise straight up at a speed of up to 2.3km/s, setting a huge trap for target satellites. Instead of remaining in orbit, most of the air molecules would fall back to Earth, avoiding the radiation belt effect and significantly reducing the risk to other satellites or spacecraft, according to the simulation.
A Chinese military study published in May called SpaceX’s Starlink communication network a potential threat to China’s national security and urged the development of capabilities to disable or bring it down.
[...] Conventional countermeasures such as anti-satellite missiles could take out a limited number of high-value targets, but losing a few low-cost satellites would not affect Starlink’s operations.
So, if you believe this Chinese research, not only can they nuke low-earth orbit sats by the crapload, but they can do so just in some specific area, avoiding long-term radiation belt effects. YMMV on the (have the cake and eat it) details of this, but for our purposes they rather obviously claim they can nuke lots of sats that way. They don't given any time-to-failure in the newspaper coverage, but I suspect it's intended to be faster than months (or else why call the radiation belt method "ineffective").
And speaking of megatons, Starfish Prime (exploding at 400km) did cause the failure of a bunch of satellites in the following months, by the old fashioned radiation belt method (although of course, it wasn't intended to do that, back then):
The weaponeers became quite worried when three satellites in low Earth orbit were disabled. These included TRAAC and Transit 4B. [...] In the months that followed, these man-made radiation belts eventually caused six or more satellites to fail, as radiation damaged their solar arrays or electronics, including the first commercial relay communication satellite, Telstar, as well as the United Kingdom's first satellite, Ariel 1.
(Wikipedia cites a NASA paper for that, so I'm guessing you and "FluidCode" will say that's fake imperialist science.)
N.B.: [according to the newspaper] the Chinese in silico experiment is roughly modelled after the [Hardtack] Teak explosion. A 1976 US study concludes (pp. 24-25) that the injection efficiency (how many electrons were trapped in the belts, relative to emitted) of this experiment was about 5 orders of magnitude lower than the later Starfish Prime, owing to differences in altitude of the explosion.
PS: There were much fewer satellites up in 1958 when Teak (and Argus--low yield higher altitude explosions) were conducted, and they lacked solar arrays, which was the main component that was damaged in the 1962 events on several satellites. In 1958, the US put up the Explorer IV satellite which measured both Teak and Argus; however it only had a planned service life of 60 days [and achieved 71]. (Some sources only mention Explorer IV in relation to Argus, but the aforementioned 1976 paper says it measured Teak too.) The Soviets had their Sputnik 3 up, which had 10x longer time in orbit, but it experienced an unrelated tape failure, so was unable to record anything of relevance from the US experiments of '58. The Soviets did use their ground stations to measure and even initially publicly report more of the Argus results than the US did, that way. (The US had a long debate how much of it to declassify, then.)
