Why is the power going on and off after I replaced the battery?

This is commonly caused by damage to the battery connector after repair. One of the pins coming off the battery carries data about the battery including including charge level.

The iPad may not be able to tell that enough power is present in the battery to power up fully.

As much of a pain as it is, it might be worth it to verify this. The pin highlighted below is the one most likely to cause this issue.

Other than that, it could be that a it needs to be left on the charger for longer, it may be struggling to get enough power to come on if the new battery was depleted, or even that the replacement battery is faulty.