Earlier quoted context omitted.
Skin in the game means participating in the community and the local economy. An investor doesn't buy groceries at the local store, doesn't go to local music venues etc. They help out the construction industry and the FIRE (finance, insurance, real estate) industry, but local neighbourhoods weakly benefit.
That's not what "skin in the game" means, and presumably these properties are being rented, in which case the renters are still buying groceries, etc.? Again, I'm not advocating for foreign investors; just trying to understand the problem.
Even if these properties are being rented out, persons with foreign capital buying power that exceeds that of locals have a distortionary and, from the point of view of a local, a negative impact on the real estate market, as they are able to out compete locals and bid up the price of housing. A person which may have normally been able to buy a home can no longer buy, and now has to devote more of their income to renting from a foreigner. Why would any local support this state of affairs?
At a high level "skin in the game" means involvement and having some commitment. That can mean monetary involvement or otherwise and I'd use the broader definition. An absentee foreign buyer has tenuous involvement in a city beyond the impact of their foreign capital. Sure they've risked their capital on an investment, but from a locals point of view they have no real personal stake in the community.