This laser focus on one specific area for investment raises the value of real estate in that area to out of reach for most of its native residents. This is where the "emphasis on it [foreign investment] being bad" comes from.
It's good for those who already own real estate, bad for those who want to.
A partial solution has already happened: the cancellation of the immigrant investor program.
You don't even need to do a higher rate - they've likely bought their properties in nicer neighborhoods at higher prices, so the property tax revenue from the is higher anyways.
Foreigners influx has some influence but not as much as one might think.
It's the sub 3% mortgage rates and 5% down special that are driving housing prices up.
Like you said, wait until interest rates hit even 6-7% and watch what happens to housing prices.
Unfortunately, when the market consists of Vancouverites making the average wage here, and also wealthy Chinese businesspeople with hundreds of millions to spend, that directly affects the market.
There is currently a condo development called Vancouver House being built in the densely-packed (for Canada) downtown core (which is also a peninsula, limiting growth and densification), which is being marketed specifically to the rich – so much so that when it was time to start shopping the properties around to buyers, the developers went straight to their target market: rich investors overseas, e.g. in Singapore and elsewhere. Even if you were an equally rich investor in Vancouver, they weren't offered here.
Vancouver House is, of course, absurd luxury, the kind that only the rich can afford to own (and maintain), and it makes sense; turn that plot of land into housing that the locals can (and are willing to) afford, and you're looking at selling for up to a million per unit. Gear the entire property towards the world's wealthy elite, and you can build units five times the size that will go for twenty times the price or more - and guarantee a higher class of owner, who will treat their property with respect (or, more likely, leave it empty for now).
Welcoming the absurdly wealthy into the economy without any checks or balances results in a rising cost of living without necessarily seeing a rise in the standard of living or in average wages. Property managers start doing renovictions (mass-evictions, allowed under local law if done for the purposes of renovating the building) so that they can upscale their clientele and double their prices. High-end shops crowd out the cheaper ones (Pacific Centre, the Vancouver mall with the highest revenue per square foot in Canada, started evicting the 'normal' stores like EB Games/Gamestop, as well as several shoe and clothing stores, in favour of high-end brands such as Victoria's Secret and Sephora).
Pushing people out of the downtown core will raise the value of the surrounding land, spurring further development. Prices there will rise, forcing those people further out, in a ripple effect. The further out you force the average Vancouverite, the more transit problems you have (as you're still dealing with the same number of transit users, but you have to provide better service in the suddenly-denser neighbourhoods as well as others), and the more traffic issues you have as more people need to drive further to get to work (which in turn causes problems for transit, etc.).
So, to answer your general question, yes, it can cause a significant problem if property developers devote all their energy to making more and more ridiculous luxury for the sole purposes of selling it to foreign investors who will leave them empty for years, while Vancouver already has rapidly rising housing prices that Vancouverites can barely afford, and when there's already a huge affordable housing crisis contributing to the city's homelessness.
Foreigners can't vote and therefore ultimately do not control that part of the process.
You probably can keeping building to a point where foreign investors begin to lose interest — but Americans do not generally like very dense housing, so this sort of becomes throwing out the baby with the bathwater. You're potentially making housing more affordable by turning the neighborhood into a ghetto.
As long as you're willing to offer a higher price, there's a monetary value the seller will accept in exchange for typical 30-day closing time required by lenders.