Would a person's prayer be considered valid if he intended to perform it in its remaining short time, but he delayed entering into it until its time was almost over or had expired, knowing that there was not enough time to perform the entire prayer?

The cure for obsessive thoughts is to ignore them and not pay attention to them. The intention of performing a prayer is not a condition for its validity. So, if someone intends to perform a prayer, and it turns out that the time has expired, their prayer is valid. If one catches any part of the prayer before its time expires, even if it is just the opening takbir, it is considered a timely performance (ada'an) according to the majority of scholars. Some say that it is not considered a timely performance unless a complete rak'ah is caught before the time expires. One should resist obsessive thoughts and not succumb to them, and not delay prayer to avoid performing some of it outside its prescribed time.