The title of Free Imperial City seems to have been exclusively conferred on cities located in the west and central portions of what is today Germany. Despite this fact, the HRE covered a significantly larger area including modern Austria, Switzerland, Netherlands, Luxembourg, Belgium, Czech Republic, parts of Italy and Poland etc. Why were there not more Free Imperial Cities and did the many prominent cities in these regions enjoy any similar privileges and autonomy even if without Imperial Immediacy?
I don't think the presupposition in your question is correct. There were quite a few of "Free Imperial Cities" outside the borders of modern Germany.
This includes cities like Bern, Basel, Lucerne, Freiburg, Schaffhausen, Zug, Solothurn, Winterthur and Zürich (in modern Switzerland), Mülhausen, Metz and Colmar (in France) or Utrecht (in the Netherlands).
(And this list is not complete for the respective countries.)
There were a few Imperial Free Cities outside of the central areas of the Empire, but as you say, most were indeed located in Swabia, Franconia and the Rhineland. These areas are classified as areas 'close to the king', in contrast to area where imperial power was much weaker.
That doesn't mean there were no Imperial cities in more peripheral regions. Nijmegen in the current day Netherlands, Bremen, Lubeck en Hamburg in Northern Germany, Besancon in the county of Burgundy, Metz, Toul and Verdun in Lorraine and Bern in current day Switzerland.
The become an Imperial free city, the Emperor had to confirm the cities privileges. Many free cities grew around royal towns, such as Wimpfen, Aachen, Nuremberg and Rothenburg ob der Tauber. Others were towns originally belonging to a Bishop, but gained autonomy and had their status confirmed by the emperor, such as Cologne, Regensburg or Worms. The closer they were to the Imperial center, the more interest the Emperor would have in confirming urban autonomy. Thus, some cities did manage to become autonomous, but never had their privileges confirmed. Untill it was sacked during the Thirty Years War, Magdeburg was practically a free city, but it never had this status confirmed.
Being a free city meant that the Emperor could tax the council or pawn the city to a nearby lord. Nijmegen was pawned to the Duke of Guelders, and never bought back. (The council continued to claim a special position within Guelders however, within the Dutch Republic.) Some towns like Magdeburg, Brunswick, Munster and Groningen were practically free, but never officially claimed this status. That meant freedom from Imperial taxes, but when their Lord was powerful enough to attempt to subjugate the city, they had no legal protection.
Which cities were counted as free was only codified between ca. 1500 and 1648. Bremens free status was confirmed even after the TYW to keep the city out of Swedish hands. That meant that some cities with a ambiguous status would never be listed as free if their Lord managed to subjugate them before this time.
Hope this clarifies things!