It is definitely not true that Discover interchange rates are significantly higher than Visa or Mastercard.

I’ve put below a list of the actual interchange rates for various personal Visa, Mastercard, and Discover cards types.

Debit:

• Visa Debit Regulated: 0.05% + 22¢
• Discover Debit Regulated: 0.05% + 22¢
• Mastercard Debit Regulated: 0.05% + 22¢
• Visa Debit: 0.8% + 15¢
• Mastercard Debit: 1.05% + 15¢
• Discover Debit: 1.1% + 16¢
• Visa Debit Prepaid: 1.15% + 15¢
• Mastercard Debit Prepaid: 1.15% + 15¢

Base credit tiers:

• Visa CPS Retail: 1.51% + 10¢
• Discover Consumer: 1.56% +10¢
• Mastercard Consumer: 1.65% + 10¢
• Mastercard Enhanced: 1.8% + 10¢

Rewards cards:

• Visa Rewards Traditional: 1.65% + 10¢
• Visa Rewards Signature: 1.65% + 10¢
• Discover Rewards: 1.71% + 10¢
• Discover Rewards Premium: 1.71% + 10¢
• Mastercard World: 1.9% + 10¢

Premium cards:

• Visa Rewards Signature Preferred: 2.1% + 10¢
• Discover Rewards Premium Plus: 2.15% + 10¢
• Mastercard World Elite: 2.3% + 10¢

You can plainly see that Discover tends to be more expensive than Visa but is cheaper than Mastercard. The only reason I could see that someone might refuse Discover is because Discover cards are all rewards credit cards that go into the higher tiers, whereas many Visa and Mastercard cards are debit cards which go into the lowest tier.