Vorschlag für englische Synonyme (im Sinne von "bewältigen"):

to cope with sth.; manage

"Zu Rande kommen" würde ich nicht als "umgangssprachlich" einordnen, sondern, im Gegenteil, als gehobene Sprachebene.

Nicht zuletzt deswegen scheint mir "mit jemandem zurande kommen" im Kontext von sportlichen Wettkämpfen - offensichtlich der einzige Anwendungsfall, wo die Wendung auf Personen bezogen ist - unpassend. Wer spricht davon, mit "dem gegnerischen Spieler zurande zu kommen"?

Two points:

“To get to grips with” is marked [fig.], while the corresponding expression with “come” is not. This is how it is in the existing LEO entries, but I don’t see any good reason to follow that example. I would say that the label “[fig.]” isn’t necessary in either case.

Also, while the German expressions – mit etwas zurande kommen and the existing LEO suggestions mit etwas zurechtkommen and etwas in den Griff bekommen – can sometimes be suitable, they don’t seem to me to cover the most common meaning of “get/come to grips with”. See the dictionary definitions given in the OP:

to make an effort to understand and deal with a problem or situation (Cambridge),

If you come to grips with a problem, you consider it seriously, and start taking action to deal with it (Collins).

to begin to understand or deal with (something, such as a problem) in a direct or effective way (M-W)

I don’t feel that expressions that mean etwas bewältigen, meistern können really convey this adequately. (But someone who is more nearly bilingual than I am might know better.)

[Edit: actually, I see that LEO does have a second translation for "get to grips with" -- etw. in Angriff nehmen. Perhaps one should add a similar entry for "come to grips with"?]

At the risk of making myself unpopular, I will say that I think there is a disconnect in the OP between the English dictionary definitions and the four examples from the media, three of which are about sport and relate to a person. (In the other one, it's not "come to grips with him" but "come to grips with him dying", i.e. with the fact of his death.) With regard to the use of these phrases in relation to sport, I lack the knowledge to comment.

Incidentally, I am far more familiar with the “get” than the “come” version of the expression. I see that M-W marks “get to grips with” as British.