I thought Rio's focus on "part-timers" was more "maybe I should be less of a leech" rather than "let's spend time with the baker".
That's just the in-universe explanation. Readers know what's up.
Although more seriously, the way Rio is behaving, doing chores, and generally being helpful, to Ruriko's surprise, makes it very clear that seeing the normal couple life of Kurumi and Ruriko has given her some inspiration to do something more with her life. Looking for a part-time job, in a bakery that sells cutely-shaped bread, seems like a good step in that direction that mixes some amount of responsibility and still doing something fun.