My kid as well as my friend’s kids all enjoyed cake and gifts on their first birthdays.
Is there some kind of allergy or physical disability preventing this? If not you should give them the experience.
My kid loved pulling tissue paper out of gift bags & playing with the bows & ribbons. They found it exciting to see new toys come out of bags & boxes. Seeing how they react to cake is always fun. One cried because he didn’t like how it felt on his hands. Another was covered from head to toe in green icing & was quite proud of himself. I get having a little one that is overwhelmed by crowds. Have a celebration at home with just you. Maybe commemorate the day by planting a tree or a special bush. Make something with their handprint. Make a special book about the day they were born & share it with them.
Buy a membership to the zoo, aquarium or botanical garden & take them when most kids are back in school.
Also kiddo is born in the early summer so the weather is usually beautiful then. We have a neighborhood park with a gazebo that you can “rent” (it’s free you just have to sign up). We hosted some close friends. Only lasted a couple hours.