The Upper Rhine was and is no Natural Frontier
The Rhine in northern Switzerland can not be regarded in any useful sense a boundary, although the river is the official frontier between Germany and Switzerland nowadays (canton Schaffhausen also covers a territory north of the Rhine). The commercial, social and political contacts across the Bodensee (Lake Constance) between German (Swabian) and Swiss cities, abbeys and bishoprics continued without major interruption from the Roman period onwards. The High Middle Ages and the Early Modern Period show a continuation of this situation until 1500.
Aargau and Thurgau belonged to the Habsburg dominions, but were occupied by troops of the Swiss cantons in 1415 and 1460. The common administration of these territories (common Lordship or gemeine Herrschaft of the Untertanengebiete) by the (sovereign) cantons, the role of the confederal parliament (Diet or Tagsatzung) and the (juridical and political) relationship with the Habsburg rulers are rather complicated. Several Swabian (imperial) cities were associated with the Swiss Confederation, for example Constance and Rottweil. Constance by the late fifteenth century was even on its way to become a Swiss Canton. It never happened.
The so-called Swiss or Swabian war of 1499 can be regarded as the violent explosion of tensions. This (civil) was was fought along the length of the Rhine, from Alsace to Arlberg, from the Upper Rhine to the Alpine Rhine in Graubünden. This war was concluded by the Treaty of Basel in 1499. The Treaty of Basel is a watershed. The Swiss cantons gave up their aspirations and ambitions in southern Germany. The main international conflicts before the Reformation between France and Habsburg shifted to Flanders, Italy and Burgundy. Thereafter, the situation in northern Switzerland did not change until 1648, 1815 or 1848.
The Rhine and some territories north of the Rhine of canton Schaffhausen became the frontier of the Swiss Confederation. The river was not a natural frontier, however, rather a topographical point of orientation than a boundary (Johannes Knebel, 1460). The modern conception of frontiers or boundaries is rather inappropriate to define territories until the rise of the nation state in the nineteeth century. Feudal lords, bishoprics, abbeys and patricians owned estates, jurisdictions, rights which transcended frontiers or boundaries in the Rhine region.
South Germany was ruled by a monarchy, the Swiss Confederation by the cantons, the diet, patricians, burghers and wealthy peasants. But even the Eidgenossenschaft counted many ecclesiastical and aristocratic powerbrokers until the end of the ancien régime in 1798. Swabian and Swiss were definitely not separated by incompatible political values and the Upper Rhine never had the (mythological) value of the Hochrhein between France and Germany (Source: T. Scott, The Swiss and their Neighbours 1460-1560. Oxford, 2017).