If I can give my 2 cents, I was in the situation you are in not long ago and I've been working in the industry since 1 year. I am not a programmer but a sound designer.
I started just like you did, by sending resumes to every big companies. Got the exact same mail as you from EA (minus the job title), got a similar one from Blizzard etc... Only Valve had a guy nice enough to answer me directly to tell me they hire only after 5+ years of experience and don't do internships (which is something I knew but I tried it anyways) and then we talked a bit which was pretty cool.
After having no luck except with one studio in LA who were interested but needed to wait to make a decision (I couldn't afford to wait for their decision) I decided to lower my expectation (my goal was to have an internship in a major studio in the US and I am from Europe so it is almost impossible) and started to search only in big EU studios. Didn't get more luck.
It also turned out the interested studio in LA didn't take me after all.
Then I started to do small studios in France (my country) and I got 2 answers, one of them I got a phone interview and I couldn't be available at the time they wanted because of school so it didn't work out and the second one is where I got my internship but I got it by being able to do more than regular sound design, I know basics in code and I know a good amount of DSP which got me the internship. Got lucky after, I got to do some sound design as well.
I got a contract there after and I've been there since 1 year not including the internship.
Random facts :
- Keep in mind that I sent probably 100+ applications and I got 4 semi-positive answers and one positive, I did get lucky especially as a sound designer (it should be easier for you as a programmer) but I had to sacrifice my original goal.
- At the end I was so desperate that I would send application to every studio that was obviously bad just because I needed the internship for my degree and I knew that internship are extremely important for the start of your career.
- My school didn't help me at all and I knew no one.
- I had a meeting with the head of my degree so he could hopefully give me a random internship in a non sound design field (my school was not specialized in sound design) and I got my positive answer 2 hours before the meeting.
All of this to show you that it is hard to get an internship and you should not give up too fast, if someone doesn't answer you and they are a small studio, just send one email to check in or call them. Small studios are usually nice. For big studios they get too much applications to care.
Don't be too cocky just because you have more than what they ask for, plenty of people have it. Having more than what they ask might put you in the top 10% of the applications but that's it. I had more than what they ask even for an entry job.
On the topic of "what they ask for" don't wait for an internship "job" to pop up on their website, just send applications everywhere.
I hope this is helpfull.