Unless I missed it along the way, I don't think anyone commented on the main reasons you were advised to maintain wings level - which is a common technique on large jets.
(a) drag with wings level is not critically greater than with a few degrees into the operating engines. Note that bank into the operating engine(s) is relevant for Vmca determination (as it reduces Vmca for the OEM - the certification limit of 5° keeps everyone honest) but is not usually necessary for continued OEI flight. In general, optimum bank angle is around 2-3° for OEI climb performance (for the reasons cited).
(b) some jets have AH/ADI which are not suitable for maintaining small bank angle indications.
(c) the 737 lateral control system includes both aileron and spoilers. If control inputs get to the stage of commanding spoiler activation then drag increases significantly. Trying to maintain a constant bank angle requires a control wheel input. While I can't recall the angle at which spoilers will rise, it is easy to get into that situation.
Overall, the compromise is to accept a very slightly degraded climb performance for operational simplicity and reduction in pilot manipulative workload.