We are planning a trip to Rhodes city for an event at the end of September and would like to spend a few days before or after on the beach. We have already visited Lindos and spent some time on Karpathos.
Any other recommendations for quiet beach hotels on Rhodes or nearby islands? We are an active older couple.

We visited Symi as a day trip from Rhodes but didn't stay there. It was beautiful and we would have liked to return. On Symi we had a few hours exploring (some people swam at the beach) and lunch in the main town and earlier, visited the stunning monastery for a short time. Worth a look to see a different island.

The problem with Symi is I think all the better beaches need a boat. I don't think it's a huge issue . There are people who take you out.

But if you're just using the town beach why not stay in Rhodes town? It has a beach. It's not the best one on the island but depending on what sort of beach you want it might be fine.

Kind of depends on how important the beach time is and what sort of beach you want

The best beaches IMO are Tsambika (accommodation above the main road or at Stegna) and Agathi (accommodation at Haraki). Agathi has neither restaurant nor beach chairs this year, however.
The walk from Stegna to Tsambika (hiking trails) is more then 1 hr OW, however. The access from restaurant Panorama (bus stop) is much shorter https://www.panoramatsampikas.gr/index.php/en/.
The walk rom Haraki to Agathi is about 1/2 hr (tiny hiking trail via Agios Nikolas or road).
The most popular sandy beach is Faliraki, with a big, noisy and rather ugly settlement (tons of hotels and bnbs) behind.
The best big package tourist hotels with sandy beaches are north of Faliraki and on both sides of Lindos.
The best sandy beach of Kos is Paradise Beach, close to Kefalos (bus from Kos town via airport)