Optimizing PostgreSQL where clause with OR
Solution:

create index on my_table (name); 
create index on my_table (other_name);


Credit: @a_horse_with_no_name

Comment: I am befuddled that I didn't find this myself. I'm sure I tested [multiple] variations of adding indexes, and I don't know how I missed the obvious one. But I'm so happy that I'm giggling, and I'm grateful for the additional eyes on the problem to find this simple workaround.