After having lived in SG now for almost 1.5 years, I think I can answer my question with a clear "It depends". This is my perspective, written from someone who is used to the medical system in Germany.
It is true that I got a health insurance from my employer (it's not like in Germany that some amount of money is deducted from my salary that goes to the state, it's a private insurance). I can't go to any doctor, I can choose from about 250 "panel doctors" that have a contract with the insurance company of my employer. Dental is usually excluded. For general practicioners I pay $5 per visit, flat. Medicine I have to pay for by my self, at least for over the counter drugs. Sometimes the doctor gives me some prescription medicine, which he has ready in some small bag or so in his office, this is for free then. I didn't need any expensive medicine so far beyond that. As far as I know there is a limit of around 15.000 or 20.000 SGD of coverage in total for medical expenses (not limited to medicine).
But I also know other expats that have much less coverage, only GP, no clinics. So yes, do check with your company and their insurance company what is included and what is excluded and how much in absolute dollars.
There are many clinics and GPs in and around SG, so it's not so hard to find one that accept my insurance card. I found that some of them are not really super clean or the doctors are especially knowledgeable, to be honest. I don't doubt that there are very high quality doctors and specialists in SG, but those are not covered by my insurance so I would need to pay it myself. In the end it's then a decision of money and "urgency". If you avoid peak hours, queuing is not an issue as far as I can tell. For a normal cold I also go to just a normal GP, but for dentists I go to some more expensive ones in downtown core. For normal dentist consultation that sets you back at least $90, without anything being done. Dentist bills can grow quite considerable, so be careful and aks first how much they charge if you don't have insurance yourself!
So to answer the question: probably you will be fine, although it will also give you some scary moments (coming from European style healthcare systems). Like, you get your first cold in Singapore, get a bit frightened, then think, maybe I should have checked that up properly, realize that you have to pay yourself, then just get some paracetamol from a GP for $5 and hope for the best.
There are many insurance companies in SG that provide additional coverage, so you can get dental coverage additionally etc. But it really depends what you want and can afford. If you only stay 3 months for an internship and are yound and healthy, go with what the employer provides, otherwise you might want to look into that. That is my personal opinion, YMMV as usual.