How true is the stereotype that PhD graduates are overqualified?
First things first - this question is very region- and field-specific, as cultural norms and views of academia differ a lot in different places. Further, this will naturally also be highly employer-specific, as not every recruiter or employer has the same sort of requirements, views and prejudices. Everybody who tells you that there is a hard and fast rule is lying.

That being said, there are a few observations that I think are pretty general w.r.t. the PhD job market:


Nobody hires you or pays you more just because you have a degree. It's a pretty common misconception that having an advanced degree somehow magically entitles you to higher pay even if you do the same job in the same quality than somebody without a degree. This is particularly pronounced in computer science, where there are a bunch of excellent programmers / software devs out there without degree. Don't expect to make more than them just because you got your training in an university rather than through participation in open source projects.
At least in central Europe, no smart person does not hire you just because you have an advanced degree. I am confident to say that in Switzerland, the sentiment expressed in your quote is entirely wrong. A PhD certainly does not close any doors. There are a lot of jobs where it will be useless, but I have not yet seen cases where it was actively bad for the applicant to hold a PhD.
There are a bunch of jobs out there where holding a PhD is in fact very good for you, even outside academia and research. Here in Switzerland and IT, consulting is one of those fields. Consultants with PhD look smarter on paper, and hence their time is easier to "sell" for horrendous amounts of money to customers. It is pretty well-established that in this case just holding a PhD will in fact give you better starting salary and better career opportunities in many companies.
Generally speaking, as a PhD holder, you will have less experience than somebody who worked in the industry for 5 years. Contrary, you should (at least hopefully) have a higher level of formal knowledge and understanding of underlying principles. What you need is either (a) a job that requires the latter rather than the former (various "evangelist" or innovation jobs come to mind), or (b) and employer that has a time horizon large enough to allow you to get experience and who has problems complex enough that they need somebody with deep fundamental knowledge. This is the reason that big tech companies such as Google, IBM, or MS, often also pretty aggressively recruit IT and math PhDs, rather than going for cheaper and more experienced professional software devs.