The white "patina" is most likely some form of opal as you suggest. I did some research and I'll explain how the agate becomes opal and why it's more common in the eroded rock. Some background:

• agate and opal are both forms of silica with the chemical formula SiO2
• opal is a hydrated amorphous form of silica (1)

You would find the opal more often on weathered and deposited chalcedony because "the polymerization of silica into three-dimensional gel-like networks [opal] is favored in the pH range 3 to 10 where dissolved salts are present in solution."(2) The pH of stream/river water ranges from 6.5 to 8.5 and all natural water has salts of some kind. (3) Therefore, dissolved silica in stream/river water reacts with the agate and opal grows onto it.

To sum up: yes, the "patina" is most likely opal and it forms when stream/river water interacts with the agate.

sources:

1. https://en.wikipedia.org/wiki/Opal
2. https://rruff-2.geo.arizona.edu/uploads/CM46_139.pdf
3. https://www.grc.nasa.gov/WWW/k-12/fenlewis/Waterquality.html

Source 2 talks about various types of opal formation, but for simplicity's sake I only mentioned polymerization.

The white "patina" is most likely some form of opal as you suggest. I did some research and I'll explain how the agate becomes opal and why it's more common in the eroded rock. Some background:

• agate and opal are both forms of silica with the chemical formula SiO₂
• opal is a hydrated amorphous form of silica (1)

You would find the opal more often on weathered and deposited chalcedony because "the polymerization of silica into three-dimensional gel-like networks [opal] is favored in the pH range 3 to 10 where dissolved salts are present in solution."(2) The pH of stream/river water ranges from 6.5 to 8.5 and all natural water has salts of some kind. (3) Therefore, dissolved silica in stream/river water reacts with the agate and opal grows onto it.

To sum up: yes, the "patina" is most likely opal and it forms when stream/river water interacts with the agate.

sources:

1. https://en.wikipedia.org/wiki/Opal
2. https://rruff-2.geo.arizona.edu/uploads/CM46_139.pdf
3. https://www.grc.nasa.gov/WWW/k-12/fenlewis/Waterquality.html

Source 2 talks about various types of opal formation, but for simplicity's sake I only mentioned polymerization.