This is true unfortunately. I am also grad student who had a similar situation and have a few tips.

• Do you have a pre-existing condition or a new issue? I have ulcerative colitis (have had it many years) and was having a severe flare earlier this year but they couldn’t get me in to GI for months. Eventually I just called Yale Health Student Health primary care and they were able to see me same day and prescribe me steroids. It isn’t a permanent fix but it restored my ability to function. You may have heard that Yale Health student health is horribly incompetent. This can be true. It really depends who you get. This approach works best if your condition is a chronic one and you know exactly what you need.
• Acute care is also an option
• Are you by chance still under 26 / on your parents insurance? If so, you could try going that route. See what’s in-network there and if you can make an appointment with GI without a referral, or if you can get a primary care Appt in network with that insurance and then GI referral through that
• If your only insurance is yale student, you’re going to have a much tougher time. It’s a PPO meaning that they only cover in network, ie, at the student health center, unless you have a very specific referral out. Confusingly, not all yale physician groups are in network with yale student health. You can ask member services about specific doctors, but you might have a very tough time finding one that they’ll cover.

Feel free to DM me if you have more questions as well