If Mythology & ancient sites are important (as well as good beaches) I share the advice to skip Hydra. People who come on this website are "Hydralized" by R. Steves' constant recommend of this island only (I think chiefly because it tacks on nicely as a day-trip at the end of his Greece mainland tours) ... but its ancient-site interest are few, and no beaches really... mainly a pretty port.
With only 9 days I'd instead sugget spending extra 1-2 days in the Nafplio area -- the Old Town itself is on a peninsula, surrounded by water, thus feels ike an island... has a cute Town beach on other side of its Cliffs, and 2-3 other wonderful beaches nearby. With a car, u can stay in Old TOwn & drive out to beaches, or even stay somewhere like PORTO ASSINI Apts ... which is directly ON a beach (it's on booking.com -- its listed as being on Tolo beach but it's actually not on this crowded UK beach). It's just 15 minutes outside of Nafplio (and next to one of my fave little ruins, Assine). Staying here is the best of both worlds... a fine beach, not overcrowded even in high season, and just a few miles away from history & scenery galore. Just check out http://www.visitnafplio.com -- wonderful NONcommercial website which has solid info/pix of entire area.
You could drive back to Athens late afternoon of your last day and drop car at airport -- and stay at one of several hotels near the airport (Hotel Avri in nearby Rafina has 24 hr free shuttle to/from) or keep car and stay at St. Thomas B & B ... do not dither about making reservations however, because you're going in HIGH season. It's "no-risk" if you take the slightly higher rate that allows free cancellation w.o. penalty, then you can breathe easy & turn your focus to reading up/prep for your trip.
Incidentally, unless you want the experience of going across the Rio bridge, that route is otherwise quite boring along the north rim of Peloponnese, & takes more time. If you headed back the way you came and turned a sharp RIGHT at Elefsina, going toward Corinth on the biggie intercity highway, it would be a faster drive.... just over the Corinth canal about 15 miles, on your RH side is exit to ANCIENT NEMEA ... only about 5 miles off the highway. Since u are not going to Olympia, this is a super "stand-in" its the "runner-up" Sacred Games site -- https://nemeangames.org -- authentically restored stadium & temple columns, your teens will dig walking thru the teams' Entry tunnel & racing from starting blocks 2400 years old (!!). This is the Heart of wine country, so at nearby village of Modern Nemea you can stop at a bottling facility & pick up some great Nemean Red. Leaving there & going back to big highway, you then immediately drive south toward Nafplio. I wouldnt try to do Mycenae that afternoon.... just get to Nafplio, wander Old Town, relax at seafront cafe for the BEST sunset you'll see in GReece. Next AM hit Mycenae 8 am & beat the bus tours... and in afternoon, find a beach.
Another tip -- you can avoid backtracking in your Argolid-area sightseeing by saving Epidaurus for the last day, when you are heading back toward Athens. Leave @ 8 am, drive up back of Palamidi for a last stunning view, then head East across the Peloponnese "thumb" to ancient Epidaurus. Don't puzzle over the acres of ruins, or its museum -- main thing is the stunning theatre -- most perfect acoustics in world they say. After that, drive up the COAST road to Isthmus & stop for 20 minutes for a photo-op of the Corinth Canal (so deep! so Skinny!).. which you cannot glimpse from the big highway. From there, on the Big Road, it's only about 1 hour to airport.