TL;DR: Yes, it does restrict free speech, but by how much is debatable.
I don't think you can draw a line in the sand to generally determine if something is restricting free speech. Consider the following scenarios:
In the cafeteria of a retirement community where residents pay for their meals, on one particular day the cafeteria decides to make the meals free. However, if you are wearing an orange shirt that day you must either change into a different color shirt, or pay for your meals that day.
In the cafeteria of a retirement community where meals are provided for free, if you are wearing an orange shirt you must either change into a different color shirt, or pay for your meal.
In the cafeteria of a retirement community where residents give their entire life savings in exchange for living there for the rest of their lives, if you are wearing an orange shirt you must either change into a different color shirt, or pay for your meal.
The expected effect of each scenario is:
- Residents that have very little money would value the free meal more than those with plenty of money. The most likely outcome is that very poor residents would not wear orange shirts.
- Residents with no money have no choice but to not wear orange shirts. Residents with very little money would probably not wear orange shirts. Wealthy residents can wear orange shirts without any worry.
- All residents have no money and therefore no residents can wear orange shirts unless they want to go hungry that day.
Effectively all of these scenarios contain some element of restricting free speech and/or maintaining control of some sort, though they differ greatly in severity and which groups are more directly impacted.
The definition of public funds isn't in question, but perhaps it is dependent on what the tax rate is that generates those funds. At a 100% tax rate where all provisions come from public funds, there is no choice one can make (scenario 3). Anything less than that is a blurred line. Some businesses cannot survive without public funds, others can do just fine.
As a side note, this reminds me of when Louisiana had a loophole which made the effective drinking age 18 years old, but when faced with losing federal highway funding, they caved and raised it up to 21 in line with the rest of the US.