Short version
In the 1940-60s a handful of university and Veterans / military hospitals in the United States (Mayo, Cleveland Clinic, Walter-Reed, Brooke Army, Fitzsimons, Philadelphia Naval, several VA hospitals, etc.) set up what the authors of the day called “hiatus-hernia clinics” or “hiatus-hernia centers”.
They were not free-standing institutions – they were simply special outpatient sessions inside an existing hospital in which every patient who had a radiographic diagnosis of a hiatus hernia was seen, treated and, above all, studied.
When it became clear that

a) the condition was very common and often asymptomatic,
b) the anatomical abnormality could not be corrected or prevented by the conservative measures that were being tried, and
c) the new disciplines of gastro-enterology and thoracic / upper-GI surgery could deal with the minority of patients who really needed treatment,

the separate “centers” were dissolved and folded back into ordinary gastro-intestinal or surgical departments.
Nothing mysterious happened; the specialty simply matured.

The historical background in a little more detail

1. 1930-40s – discovery phase
   • Routine barium-swallow X-ray examinations, first performed in 1909 and popularized in the 1930s, started to show the sliding (type I) hiatus hernia.
   • By 1940 the lesion had been described in thousands of patients (Akerlund 1926, Allison 1948).
   • Because many of those patients also complained of “dyspepsia, chest pain, dyspnoea, palpitations, nervousness”, hiatus hernia was suspected of being the cause of a wide variety of complaints, so the condition acquired great clinical interest.

2. 1945-mid-1950s – the “hiatus-hernia clinic” era
   • Large military and veterans’ hospitals set up disease-oriented research clinics.
   – Example: “Hiatus Hernia Center, Fitzsimons General Hospital, Denver, Colorado” (McLaughlin & Harper, Mil Med 1954;115:123).
   – Example: “The Hiatal Hernia Clinic of Walter Reed Army Hospital” (Grossman et al., Ann Surg 1956;144:308).
   • The purpose was to gather big series rapidly, follow them prospectively and test both medical and surgical ideas.

   Usual conservative treatments tried in those clinics
   • Head-of-bed elevation and postural rules (no heavy bending, avoid recumbency after meals).
   • Weight reduction.
   • Antacids / belladonna and phenobarbital mixtures (the era before H₂-blockers).
   • Abdominal binders.
   • “Pneumatic reduction”: a stomach tube was passed, a syringe used to alternate insufflation and suction to move the gastric cardia below the diaphragm; radiographs showed that the hernia reappeared as soon as the patient stood up.
   • Breathing and diaphragmatic exercises.
   • For para-oesophageal hernia with obstruction or volvulus, open repair or gastropexy was done, but very few centres could offer that operation regularly.

   Results that became evident by the mid-1950s
   • The conservative manoeuvres almost never produced a durable anatomical cure.
   • Symptoms that improved usually did so because of acid neutralisation or lifestyle measures, not because the hernia had gone.
   • A great many hernias were entirely asymptomatic.
   Therefore the cost and manpower needed for a dedicated “clinic” were hard to justify.

3. Late 1950s-1970s – integration into mainstream care
   • Rudolf Nissen’s fundoplication (first performed 1955, published 1956) and later the Belsey Mark IV (1961) and Hill posterior gastropexy (1967) offered reliable anatomical repair for the symptomatic minority.
   • Cimetidine, the first H₂-blocker (1977), and omeprazole, the first proton-pump inhibitor (1988) allowed powerful medical control of reflux symptoms in the majority.
   • By that time every major teaching hospital had a Gastroenterology service and an Upper-GI / Thoracic Surgical service; a separate “hernia clinic” no longer added value, so they disappeared from hospital directories and from the literature.

What those “centres” achieved
• They mapped the natural history of sliding and para-oesophageal hernias.
• They demonstrated that symptoms are produced mainly by reflux, not by the mere upward displacement of the stomach.
• They gave the first large prospective comparisons of conservative versus operative treatment.
Their work was indispensable, but once the answers were known there was no need to maintain stand-alone clinics.

So, the short answer to “what happened to all the centres for hiatus hernia?” is:
They served their research purpose in the 1940-50s and were then absorbed into ordinary gastro-intestinal and surgical departments once the condition was better understood and could be managed routinely.