Yes, it is that simple. I keep books for a small business as a sole proprietor and applied for the Chase Ink last December. DH and I are headed to Hawaii in November with bonus points to spare.
AFAIK, from an accounting perspective, there is nothing illegal about using your company credit card for personal purchases, but you can not deduct those personal purchases as business expenses for tax purposes. So, let's say Joe Schmoe owns a bike shop as a sole proprietor. He has a business card and sometimes uses it for business, but sometimes he buys a gift for his wife. (They have joint accounts and this is the only way he can surprise her!). When he accounts for the personal charge, he will debit owner distributions, and credit the card liability. See, nothing recorded as a business expense. As opposed to buying supplies for the shop, in which case the transaction would be to debit the expense and credit the card liability.
IDK about Chase's agreement terms, but they are very unlikely to investigate, IMO. From what I've seen in travel hacking forums, my understanding is that many, many people open business cards and put personal purchases on these cards to meet minimum spend.
So, in summary, it is against Chase's terms, but it is not illegal. Some would say commingling business and personal purchases in the same account can lead to confusion or disorganization, but for most sole proprietors... big whoop. Just account for it properly and file your taxes accurate. My guess is Chase has bigger fish to fry than Joe Schmoe who got a bonus and only $500 of his $5000 minimum spend happens to be business expenses.