It's a bit more complicated than that. US agricultural policy subsidises corn production (driving prices down), but US trade policy keeps cheap sugar out of the country, while US energy policy diverts corn into making biofuels, both of which drive prices back up. It's not clear what the net effect is.
Even the impact of corn subsidies alone does not seem to be enormous, however. Some estimates have corn subsidies driving the cost of corn down by ~27%, with flow on effects to the prices of meat, dairy, and yes, HFCS. However, HFCS make up ~3.5% of the cost of a soft drink, and the cost of corn is less than half the total cost of making HFCS, which means the total impact of corn subsidies on the price of a soft drink is a couple of pennies.
In other words, if we wanted a sugary drink tax to counteract the impact of corn subsidies in the US, it would probably need to be on the order of 1-2%. Would a 1% tax have any measurable impact on consumption? Unlikely.
Meanwhile, Australia has no subsidies, mostly uses sugar (and not HFCS) but has the same obesity patterns as the US. Analysis of obesity and agricultural policy across countries finds no trends.
Source: https://grist.org/article/farm-subsidies-bitter-and-sweet/
Also, small point:
> This had the side effect of sugar being a cheaper food ingredient than most of the alternatives, even in countries that didn't directly import HFCS (since the US was no longer removing as much raw sugar from the market, driving down costs).
Only in the short term; that effect would have ended years ago. The global price of sugar now is driven by the marginal cost of production in Brazil, mostly. Americans don't consume (much) imported sugar, but they also don't export much HFCS, and most of what they do export goes to a single country (Mexico). There's no glut of "sugar the US would have consumed" on the market.
Second, global sugar prices have been pretty flat for the last 20 years; I don't think your link 3 is really showing what you say. You can poke at the chart here: https://www.macrotrends.net/2537/sugar-prices-historical-cha... but I'm not sure what I'm meant to be seeing.