In my experience, the credit card services you talk about are actually quite useful, a concierge will find you an excellent price, the trick, do not use their web site, call them, the rewards will apply the same, but the concierge will get a similar deal than the one you may find in services like google flights1 or Skyscanner2 or Hopper3. Just be aware, the call to a concierge and their service will have a cost, check with the credit card,depending on the card you have could be waived, it is not much, last time I checked it was around $35.
Find the best fare you can, write the details down and call the concierge, tell him you want him/her to improve it or match the offer, they may surprise you, in my experience, at least 3 out of 4 times they will at least match it or will be so close that is worth it.
WARNING
You may find some shady offers in skyscanner and hooper, use the filters to remove the low cost airlines and select direct flights if applies
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