First, I'm so so sorry for your loss. It's so easy to blame ourselves in hindsight, I think many of us have suffered losses that felt preventable, especially with such fragile prey animals as birds, and it's so easy to point the blame at ourselves because it feels like someone must be at fault... be kind to yourself, if your best friend lost someone close to them you'd never point at them and accuse them of causing the awful traumatic thing they're going through. Treat yourself like your best friend and be gentle. There's time to learn from our mistakes after we tend to our wounds.
That said, I do think grief counseling can be very, very helpful, and generally encourage people to try it, but as a therapy veteran I also know it is very important to find a therapist who clicks with you. The wrong therapist sometimes not only feels unhelpful, they can sometimes make things worse. I went to my school therapist in college after the death of my beloved rat Hamlet and felt WORSE after the session--she spent the full hour trying to focus on previously diagnosed mental illness, downplaying the effect of my pets death on my well being and at one point even making a comment about how "gross" rats are... I do think that experience wouldn't have gone exactly the same if my loss had been with an animal that doesn't have the same kind of hygiene stigma that rats have, such as a bird, but regardless I have a strong feeling that this particular therapist had no way of empathizing with pet loss. It's probably not fair of me to say this but she should find a different job lol.
The first session with any therapist should not be about getting down to brass tax and doing therapy. It's an interview. They'll ask you questions to get to know your circumstances, you can ask them questions to get to know their values, and if you get ANY weird vibes there's no shame in saying sorry, this won't work, and finding a different therapist. There's a ton of resources online, such as
https://www.psychologytoday.com/ that index therapists and allow them to select specialties so that you can filter your results. There's also no reason not to do a bit of discovery in the initial email--for instance, saying, "Hello, I'm looking for a grief counselor who is specifically comfortable supporting me through pet loss where I feel responsible, are you taking new clients and comfortable with this and if not do you have any referrals?"
Best of luck to you