@jeanmandredeix
The short answer is no. There's no requirement to be part of the public health system just because you're a resident. However, depending on your residence status, you might need proof of health insurance when you renew, and this can be either NHIF coverage, or a private health insurance policy, or the immigration-oriented Health Insurance for Foreigners policy.
On insurance policies, I'd point out that the above mentioned Health Insurance for Foreigners is cheap (100 euros ish) and purchased just to meet the immigration requirements. It can't be relied on as a true private health insurance policy, these are significantly more expensive, especially if you're older and/or have pre-existing conditions.
If you can afford private medical insurance, then that's probably the luxury option.
However, I'd say public health coverage (in NHIF) is a fantastic deal, and I recommend it highly. You can have both public coverage and a private policy, or just one (or the other). I pay into NHIF as a voluntary contribution (unemployed/economically inactive) and it's around 30 euros per month. I'd say the only disadvantage is that once you opt in, it seems quite difficult to opt out and they'll expect you to maintain your payments indefinitely. On the other hand, that 30 pm is far cheaper than a true private health insurance policy.
Once you're in the system, you will need to choose a GP, just like in the UK. I wouldn't surprised if, supposedly, GPs are supposed to examine new patients, but I doubt it's enforced, or taken particularly seriously. The only requirement is that you choose a GP and there are some rules over when/how you can change him/her. But once you have a GP, it's entirely up to you when (or if) you ever go see him/her.
Full coverage (i.e. from up to date contributions) entitles you to free or nearly-free treatment from your GP and public hospitals. Just like most national health systems, typically your GP is the gatekeeper, so if you want the cheap x-ray or cheap cardiologist consultation you need a referral.
With or without a GP, you can go direct to other GPs or specialists (I recommend SuperDoc as the way to find these docs online, check their reviews, and book appointments). However, you'll then have to pay whatever their full rate is (unless you have private health insurance).
Finally, there's an unusual feature of the Bulgarian public health system which I have not seen anywhere else, and this is what makes the NHIF monthly contribution such a great bargain (IMHO). In the Bulgarian system, many (most?) private clinics/hospitals are affiliated with the public system and get reimbursed by it. This means you can get private hospital treatment (from a GP or specialist), pretty much whenever you want/need it, and the cost will be heavily subsidized by the public system (i.e. you get VIP private treatment for a fairly minimal private hospital surcharge).
If you're familiar with the NHS, for example, you know that many folks with aching hips/knees find it quite time-consuming to get through their GP to get appointments in hospitals firstly for scans, and then secondly for a specialist to review the scans. Even IF the specialist agrees your aching hip needs fixing (and there's a high chance he won't), you can spend months (years?) on the NHS waiting list for the relevant operation. It will be free, when you get it, but you might wait a long time. If that frustrates you, you can go private and get your new hip next week... but you will pay FULL PRIVATE PRICE (e.g. 15k-20k I'd guess in a good UK private hospital).
Here in Bulgaria, you can go directly to an orthopedic surgeon in a private hospital and get a consultation and relevant scans in days. If he agrees you need the new hip (and he'll largely be guided by how much pain you have, and how much YOU want to have the op) then you can schedule the op whenever you want. I saw orthopedic surgeons in large private hospitals in Sofia (one was Tokuda) and all agreed I could have a THR whenever I wanted. The only thing I checked was whether they were affiliated with NHIF (otherwise you pay full price). I asked them to quote me 2 prices: the price for a standard THR as recommended by NHIF, and the price for their top-rated prosthesis with private room and a week of physiotherapy. The former was like 500 euros, and the latter maybe 1,500 euros. This is a few years ago, so the price is higher now, but you get the idea. The same thing happened when I went to the specialist (private) hernia clinic in Sofia. They checked online for my NHIF coverage, and then confirmed that my VIP private surcharge would be 500 euros. I didn't go ahead with either op, but both hospitals looked clean and modern and the surgeons seemed very credible and competent.
You should note that if you have this NHIF coverage, then any private health insurance policy that you buy should cost less (as they'll know that they'll pay less for your treatment).
My GP is wonderful, he's based in one of local private hospitals (Medline in Plovdiv). He's very low touch (i.e. he doesn't bother to chase me on anything), but he's super helpful when I do decide I need something. He has clinic hours every weekday, and unlike the UK, there's usually a very small line (3-4 people tops, and I can be first in line if I arrive 15 mins before clinic opens). He doesn't clock watch or have designated 10 minute appointments, he spends as long as needed to figure out the problem and deal with it). If I need a scan/test or specialist consultation, he does the referral straight away, and usually I've done it at Medline during the same visit (occasionally a couple of hours later, or next day).
I have been to public hospitals here (e.g. St George in Plovdiv) and, just like the UK, they are extremely crowded and you can hang around for hours. Pretty much I just go to private hospitals (Medline, Kaspela, Pulmed in Plovdiv or Tokuda and Vita in Sofia) so I've pretty much had the VIP treatment almost exclusively.
I should note that consultations/scans/tests are really pretty cheap here anyway, so I find it's hardly worth the aggravation of bothering my GP in order to get a referral. e.g. full blood tests at Ramus Labs are like 35 euros (+ a bit more for your extras), an x-ray is around 50 euros, an MRI maybe 200-300 euros (depending on what you want to scan), and a specialist consultation is typically 50-100 euros (but some might be higher, of course).
Overall, I find my medical treatment (including GP access) is far easier/quicker/better here than in the UK. I know others disagree, but my experience here has been excellent.