Exactly the problem.
I remember when I first found out the FAA accepted the cross2downwind (my term) at pattern altitude entry. It was one of those WINGS days with multiple seminars. One of the speakers did an excellent scenario-based presentation on how the course reversal to 45 is the most dangerous pattern entry. I thought to myself, "I know he's right but it's the only one the FAA approves of!" Found out a few minutes later the cross2downwind entry (used in many other countries and actually the one I learned at my base when I was a student) was finally FAA-sanctioned.
The presentation used a very common scenario with three airplanes. #1 was doing pattern work and on crosswind. #2 was approaching the pattern from the upwind side. #1 continued on an unnecessarily wide downwind (that never happens
) and #2 did that FF steeple descending teardrop. (#3 was not really a factor)