You've hit on a real problem in the games industry. We don't know how to judge game-making expertise. Training someone up to be productive on a new game project can take weeks on its own, so we're very skittish about making that investment. Our best guess is to look at games someone has already made, which leads to a catch-22 situation where you need to be hired to work on a game so you can have your name on a shipped game so you can get hired.
Going through the job postings for my studio for example, nearly all of them ask for 5 years of prior game experience and multiple shipped titles, and there's a lot of competition for the few student/new-grad level positions. This is especially prevalent at the big AAA companies. A lot of us were able to get our start because a smaller indie studio took a chance on us (or we took a chance starting our own indie studio).
Just to add to these troubles, of the studios that do hire students/grads/interns frequently, some have a reputation for churn - treating these beginners as cheap replaceable labour to work till they burn out, knowing there are always more out there to take their place. So make sure you read up on the work culture at any studio to which you apply.
There is a way to break this catch-22 though, and it might be perfect for the summer experience you describe. It just requires broadening the definition a little from "work experience" to "game development experience"
We do these things called Game Jams, which are a bit like a hackathon or charette: a group of game creators get together and decide to build a new game from scratch in a very short amount of time - often a week, a weekend, or even just one day. They're usually free or very cheap to participate in, and don't require any CV review to get in.
I'd particularly recommend jams that happen at physical locations like schools or coworking spaces, since they give you the opportunity to mingle with fellow creators, get inspired, ask for help, offer to help, and generally just ride the energy of all these people getting together to create.
There's also a continuous stream of online-only jams - itch.io catalogues many of these in a handy timeline format. If you can't find a suitable jam near you or starting on the timeline you want, you can always grab some friends and start your own. :)
Most jams will have a theme or challenge of some kind to spark your creativity, others will have constraints like using specific platforms / tools / genres. Most in my experience aren't a judged competition - they usually end with an arcade or showcase where everyone gets to play all the weird little games the participants came up with.
You can join a jam as a team of friends, or as an individual if you want to do it all solo. Some of the bigger jams also provide matchmaking services to help you assemble a team, or "floater" roles where artists/sound designers without a team get paired up with teams that lack artists/sound designers.
I recommend jams because they reproduce a microcosm of game development - all the same pressures and joys and frustrations, team dynamics and scope challenges (and oh gosh the bugs!) condensed into a low-risk opportunity. If a jam goes badly, ehn, I lost a weekend at worst - I didn't have to live through a studio I founded going bankrupt. ;)
And the games that get created aren't just throw-away. With a bit of dedication and polish they can become great portfolio pieces to bring to a prospective employer to prove you have game-making skills. Here in Ontario, we've had a number of success stories of creators taking experimental game jam prototypes and continuing to build them up into finished games you can buy on Steam/Xbox/Playstation/etc, including Runbow, Keep Talking and Nobody Explodes, Toto Temple, Mount Your Friends, Super Time Force, Lovers in a Dangerous Spacetime, The Yawhg. (I have no affiliation with any of these games, I just think they and their creators are pretty awesome)
So yeah, all that to say: game jams can be a fun and accessible way to build up gamedev experience, sidestepping industry gatekeepers, and I highly recommend them even for experienced game developers as a way to periodically stoke your passion for making games. Feel free to hit me up on Twitter if you want more information about jams. :)