Welcome aboard, Sugars. My first kit was a 350D and a 70-200 F4 L. Very similar across the board, with 8mp and a very sharp 200mm focal length. I was budget restricted at the time, so I made do with these for a while, forcing me to work on my field skills. Getting close is a challenge with run and gun style shooting, but a 300mm lens can give you good results from a blind. I have had great success through the years by placing a feeder and some low lying perches outside of a basement window. That's just one approach to help you think about your options.
If you really want to upgrade your kit I have some basic suggestions. First, I think the 400D is a 10mp sensor, IIRC. With my 350D and 20D bodies at 8mp, I never felt resolution limited. But that was 15-20 years ago. Most computer monitors and web pages operated at lower resolutions back then, so we didn't know any better. The improvement in resolution since, with everything from computers and TV screens, to smart phones and iPads, means those 8-12mp cameras show their limitations now, especially with cropping which is common with bird photography. That said, if you can shoot from a blind, and get sing birds on a perch 2-3m away, you can catch some terrific feather detail. If you want a kit you can take to the local park, I think 20mp is a better starting point.
Next, a better lens will most certainly improve you images, especially if you pair it with a higher res sensor. Tim's recommendation for a 400mm F5.6 L is dynamite. I'll add that I owned the sister lens for a while, the 300mm F4 IS L. Both are incredible, and can be had for under £400 on the used market in excellent condition.
If I was looking to upgrade from a kit like yours, right now, with my current knowledge, and a reasonable budget, I would go to the used market and grab a Canon R10, an EF-RF adapter, and a 300 F4 IS L. But that's a personal bias toward gear that I'm comfortable with. For the record I Currently shoot with an R7, EF-RF adapter, and a 400 F4 IS DO. Check out my gallery if you're curious. Most of my work since 2022 is with a 70D or R7, with the 300 F4 or 400 F4.