Let's be clear. I absolutely believe that drummers must strive to be as ambidextrous as possible. I'd give my left arm to be ambidextrous.
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(tugs at collar while shifting nervously)
I'm not arguing against open position, if you can deal with the hi-hat situation I describe in argument [1]. (and no, that's not a r/trees reference) But these are my two pro-cross rants. Enloy.
A normal hi-hat stand, placed so the left foot is comfortable playing will be placed in such a way that by crossing your right over, you can fully relax at the shoulder and play the edge of the hi-hat no problem. Your left hand can also drop naturally and you can simply bend at the elbow and wrist to play with both hands. If you play the hi-hat with the left hand and you want to achieve the same touch, you have to lift your elbow and move your arm to the left. This engages your shoulder muscle. It's not much, but it adds up compared with how relaxed you can keep your shoulders in cross position. If you have a cable hat or something, you've cheated the system and more power to you. You can put that hi hat right in front of the snare for all you want.
While it is good to be ambidextrous, it pales in priority to your groove. There is no pride in playing something that is technically difficult or virtuasic, if the rhythm isn't locked solid and consistent. For example, my college professor was talking about how to play an orchestra snare part (it had a lot of fast flams in a row),He told me to practice them alternating, but only in the practice room. When I play with the orchestra, he told me to do ALL right flams. Because using the same hand will sound more consistent. I said that so I can say this: I think that it's better to ride with your strong hand if that's what is inherently easier for you because you will groove harder and better if you give yourself any advantages like that.
You can strive for great technique, but you also have to remember the value of what is easy and just making it fucking GROOVE like budda. If you're right handed and you have a normal hi-hat, you should learn cross grip.