you should be able to manually boot into recovery, bootloader and download modes. If your phone is powering on when the battery is inserted, it means there is a positive signal going through the power button to complete the circuit, in some way. With that said, let's assume you inserting the battery is the moment you would hypothetically be pressing the button if it worked. So in this individual case of the battery insertion acting as a power button depressing, it should be possible to get into any one of the modes desired by keeping any additional buttons depressed while inserting the battery.
In other words, to get into download mode press the vol up+down buttons at the exact same time the battery is inserted. it should work. My sister's evo4g had the exact same symptoms your phone does, and it worked for hers. I was easily able to boot into the bootloader, as well as recovery by following the steps I mentioned, and since it was a touch-based recovery, I was still able to flash/backup/restore, etc...I had full control over recovery, and everything worked just fine.
on second thought, download mode probably won't work with our device since you have to release the power button or it will fully boot....but fastboot and recovery should definitely work!!