There are other providers besides Mastercard and Visa. Discover, Capital One, and American Express, to name a few.
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In my own haste, I had not fully considered the extent of other payment processors, so thank you for your reply. Looking over my response with fresh eyes, I am not much proud of it, but I have decided to leave it posted for the sake of discourse rather than delete it.
I am aware that some businesses (large and small) only accept Visa or MasterCard (source), and that some institutions (like my local credit union) seem to only issue Visa, which is why I expressed hesitation. However, if migrating to one or more of those alternate providers is a viable option for you, then that is great, and I encourage you in doing so.