I didn't say the doors were closed, just that the OP would have a good deal of ground to make up by unconventional channels. I don't think it has anything to do with inclusivity, as much as just pragmatics. There is a certain foundational set of knowledge and supervised experience that is thought to be needed to be competent. Most people gain this through grad school. It's more efficient. It can be done later, but the infrastructure isn't really there. As for when you get this, I know a lot of people who decided later on in grad school that they wanted this, and they were able to gain the appropriate experience to make it happen on internship and get a good postdoc. I'd disagree about the two-tiered system, personally. It should be easier for those who specialize earlier. Those who specialize later can still do it, they just need to make up ground. That is the consequence of their decision.
I am one of those people, though it took a lot of extra effort, strategic planning with my (non-neuro) mentor, and a bit of luck. I didn't really consider a neuropsych path until my 3rd year, though I had spent my first two years conducting research w a heavy emphasis on peds assessment. I also spent 6 months in the on-campus assessment clinic (through the research group) doing peds assessment. I didn't have a neuro mentor and I couldn't get into the neuro practicums bc they were highly competitive and I didn't have the required classes to apply for the neuro clinics in my 3rd year. I was able to take a number of classes outside of the program (Neuroanatomy I&II, Neurophysiology I&II, Organic Chemistry I&II), as well as take a more advanced stats class than was required. My fourth year was spent in a non-neuro practicum, taking more classes, and finishing my research to defend. I later enrolled in a campus-based Pharmacology program and then took a (5th) gap year to finish those classes while I continued to do research, TA, and apply to internship sites.
I did my internship site ranking based on who would be the best neuro mentor fit for me. I matched to a generalist program, though I worked primarily with an amazing neuro mentor and basically negotiated with my DCT to get as much neuro assessment experience as possible. I had other rotation supervisors with neuro training who were willing to work with me on cases. I was at a large VA hospital who happened to get a large influx of OEF/OIF veterans who needed TBI assessments, so I got lucky in some regard that I could take on additional neuro cases and also do TBI related C&P evals. It was a little rocky at times because I had to complete my regular internship duties (like therapy, ugh) and also squeeze in a ton of supplemental reading of articles. Since I had defended prior to going on internship, and the rest of my internship cohort had not, I was able to use the protected "research" time each week to review case studies w my mentor, who also taught me how to review imaging studies and interpret imaging reports.
Applications for fellowship spots started Dec 1st, so I tried to get as much setup in my rotations as I could, so I could talk about what I was doing to add to my foundational learning. I secured strong letters of rec. I benefited from personal connections I had made over the years for mentors who could vouch for my abilities, even though I didn't have a traditional training path for neuro. Connections matter in neuro, for better or worse. My pharma training added another dozen classes to my experience, and eventually I was able to secure a highly-competitive fellowship spot. I share all of this not to brag, but to demonstrate that it took *a lot* of extra work. I still had some weaknesses in my training that I had to address during my fellowship years, but I also had some strengths that made aspects of my fellowship easier. I still felt behind at times because I just didn't have the volume of neuropsych evals that the other fellows had, and my exposure to some adult measures was less. I did have other knowledge like more exposure and experience w peds assessments & associated research, so during the Adult & Peds combined didactics I could offer more. I also developed an excellent grasp on neuroanatomy and had a lot more knowledge about pharmacology, so brain cutting didactics w the pathology residents and team rounds were times when I could excel.
To the OP, you are definitely in a challenging position. Wis gave good advice about reaching out the the board and inquiring. They can best speak to competitiveness and what other options you may have if you aren't competitive for neuro-match. Depending on your other interests, maybe Rehab Psych or Health Psych might offer some of what you want, but they aren't the same as a more trad neuro career and I'd advise being pretty conservative with the types of assessments you consider offering. Pre-surg evals aren't heavy lifting if you know the supporting research, though I wouldn't suggest trying to do full battery assessments and doing everything soup to nuts bc you won't know what you don't know.