This would be an act of war and open their birds to retaliatory strikes. Tracking down the receivers and levying draconian punishments on those caught would do the trick. Moreover, from the example of Chinese students abroad, it's unclear access to information would do much.
Not by the thousands, short of taking ALL satellites out.
Kessler syndrome takes care of them all once you generate enough space junk.
The issue would be whether they could cause enough damage to make the orbital plane (temporarily) unusable.
I also think that both countries could launch geosync satellites or using existing geosync satellites to just generate a shitton of noise on whatever spectrum Starlink would use.
Starlink of course won't get a license for spectrum in either country.
The probably is there for a reason though, the exact extent of their capabilities is unclear.
Here's a citation: https://thediplomat.com/2022/06/chinas-directed-energy-weapo...
I would not call it rumors. If you'd like a more authoritative citation, here is a DIA (Defense Intelligence Agency, i.e. US military) report cited in the article I cited above: https://www.dia.mil/Portals/110/Documents/News/Military_Powe...
> During the past two decades, Chinese defense research has proposed the development of several reversible and nonreversible counterspace DEWs for reversible dazzling of electro-optical sensors and even potentially destroying satellite components. China has multiple ground-based laser weapons of varying power levels to disrupt, degrade, or damage satellites that include a current limited capability to employ laser systems against satellite sensors. By the mid- to late-2020s, China may field higher power systems that extend the threat to the structures of nonoptical satellites.
The discussion of current capabilities is primarily about optical satellites (though it can be read to include other components), but it probably applies to giant sensitive phased array antennas too.