In the realm of empty-hand combat, true ambidexterity is a rare bird. Marvelous Marvin Hagler took great pains to develop facility as both an orthodox fighter and a southpaw, but we could easily see where he stacked his chips when the heat was high.
Seeking ambidexterity is a worthy goal — but one that has "opportunity costs." That is, the time we must put into developing the off-hand (or off-foot for kicking or off-side for grappling) means less time can be invested in moving the competency needle closer to mastery on our good side.
We should always make that cost-to-benefit analysis in our training because there are only so many hours in the day and precious few hours per week we can dedicate to training itself. With actual training time at a premium, do we want so-so returns or better-than-average returns?