The biggest (but still temporary) ding you'll see on your credit score from opening a new account is from the low average (and low minimum) account age. This will have a stronger effect than the hard pull of the credit report, which is still a factor (but not much of one if you only have 1-2 pulls in the past couple years).
Having a lower average account age increases your risk to lenders. Your average will go up by one month per month, and each time you open an account it will suffer a drop proportional to the number of accounts you already had open before. So if you want to have a more "solid" credit score that stays strong in the face of new accounts in the future, it's better to open a few more accounts now (assuming you can ride out the temporary drop in score and aren't planning to go e.g. mortgage-shopping in the very near future).
Having an additional line of credit will also likely cause your credit card utilization (total balance / total credit limit, expressed as a percentage) to decrease, which would tend to increase your credit score, counteracting the age factor, unless your utilization is already extremely low (which it probably is given your monthly account payoffs). There are various credit score simulators out there, from places that show you your credit score, and you can put in a hypothetical new card account to see the immediate likely impact for your particular situation.
You identified other costs, such as risk of fraud and fees. You should check your statements once in a while even if you're not using the card, just to make sure no one else is. The bit of additional time required for this is a nonzero cost of having an open credit card account. So is the additional hassle of dealing with having the card stolen etc. if you carry it in your wallet and your wallet's stolen.
If you have an account with zero activity for some number of years, the bank may close it automatically and that can reflect negatively on a credit report (as a bank closure of the account, the reason is often obscured). Check your terms and conditions and/or have some activity every so often to prevent this from happening.
Some of the otherwise most attractive credit cards have monthly or annual fees, which will cost you, and you won't want to close those because it would then reduce your credit score (e.g. by reducing the total available credit and increasing your utilization percentage) - so the solution is don't apply for credit cards that have monthly/annual fees. There are plenty of good cards without those fees.
With a credit score that high, you can get cards that have some very good benefits and rewards programs, as well as some with great introductory offers. Though I'm not familiar with details of Amazon's offer, $80 cash up-front with nothing else seems unlikely to be among your best options. I would think that for at least some of the fee-free cards available to you, the benefits exceed the costs, and you could "cash in" some of the benefits of your good credit record to get those benefits (i.e. this is one of those things you work hard to build good credit for), while also building your long-term reputation for repayment reliability.
Also be aware as you shop around for cards that credit card companies pay fairly high referral fees to websites that send customers their way, so if you want you can think about who you're supporting when you click the link that takes you to an application you complete, and choose to support a site you think is providing a useful consumer-focused service.
As factors affecting your credit score in addition to payment history (i.e. making regular payments as agreed on the new account will help you), Equifax lists:
- Length of credit history. In general, a longer credit history is
better and can sometimes have a positive impact on your score. Credit
history typically accounts for around 15% of your score.
- New accounts. Opening multiple new accounts in a short period of time
may negatively impact your score.
- Inquiries. Whenever someone else gets your credit report -- a lender,
landlord, or insurer, for example -- an inquiry is recorded on your
credit report. A large number of recent inquiries may negatively
impact your score. Your new credit accounts and inquiries generally
make up about 10% of your score.
- Accounts in use. The presence of too many open accounts can have a
negative impact on your score, whether you're using the accounts or
not. This activity usually makes up approximately 10% of your score.