The big difference between the reality of a state etc. and the reality of the self is: Some people (nationalists, patriots) may wake up in the morning with the sense, sentiment, feeling of the nation, building identity, actions, communication etc. around it. What dictionaries say, if it’s constructed or not matters little. It is experienced as a reality.
But all people (except fully liberated ones of course) wake up with the sense of self, even Buddhists who’d like to wish this away by repeating Buddhist not-self formulas.
If we could watch a flow-chart of mental processes and replace words by symbols this would be obvious. Self-conviction is self-referential around main clusters, for example how concerns, thoughts, actions, feelings etc. are centered around the body. When thoughts and feelings come back to the same clusters or complexes there is a self-reality at work.
What is the value for this self-referential processing in saying “oh no, this is just constructed”, or “the Buddha said there is no self”? There is only value if these sentences are embedded in a larger process which eliminates the self-referential reality. In fact, these slogans are completely pointless in themselves. What matters is only if there is a larger process to reduce and eliminate self-referentiality - no matter which slogans or beliefs are attached to this process.
Hence the interpretation of the Buddha as a teacher who doesn’t care about making ontological statements per se. As a liberator he would design statements and actions leading to liberation - he would not be interested in a description of ‘reality’ in and for itself (which ontological statements are).