It's a highly individual answer.
As a heavy travel hacker, I've tended to have many cards at a time based on sign up bonuses. I've been working to get under chase 5/24 rule for the last 18 mos so have slowed down a lot and now confront the question of keeping cards more than before. Anyway I don't want to get lost in the too extreme stuff I've done/do that others wouldn't.
I personally travel enough and have other techniques that I can cash out the $200 Uber and $200 flight travel credits easily so the net fee on the personal platinum is down to $150 for me. Same goes for the chase sapphire reserve (CSR) credit of $300/yr so that's also a net annual fee (AF) of $150 in my eyes. Others discount less for these features and that may be appropriate if they can't easily and reliably cash these out. Anyway, I see each as a $150 af card to me.
Before anyone says "Spendypants" I'll just note that we have extreme credit card spend due to business activities and occasional manufactured spend.
We also use the chase ink cash card (no AF) to buy mobile phone, internet, and Amazon credit at Office Depot all at 5pts per dollar. These probably total $400/mo for me so 24k pts/yr. And we have two chase freedoms (no AF) that we max out the quarterly 5pts/dollar up to $1500/qtr in rotating categories on about 75% of the time so that's 45k chase points per year. Between these two sources, we've got 67k chase points per year whose value jumps from $.01 each to $.015 or better if we have one of the premium cards such as Sapphire reserve or preferred. That boost in value is $335/yr for having a premium card easily justifying the $95 AF on a card like the preferred. Discounting by that, the csr premium is down to just $55 ($450-$300-$95)
For business reasons, I have high restaurant and parking and transit spending which easily make up the $55 remaining fee via the 3pts/dollar on these from the csr over alternatives.
We find the chase card set critical to support our heavy southwest flying and occasional united use via point transfers. Hyatt is strong too as is Singapore and Korean if you do into. British airways can be a good one on occasion. We also use them at $.015/pt for paid tickets pretty often.
Bottom line for me is that a chase sapphire reserve plus other chase cards is a great point earning machine in significant part because of meaningful spend associated to our businesses. It's complicated but the math works. The travel benefits themselves are things we hardly use because they duplicate things elsewhere. Primary car rental insurance is top of my list that does favor the csr. Priority pass lounge access is getting better in the US but duplicates other means of access for us and Citi prestige is better there but beyond the scope of your question. Anyway, chase card set is all about the high earn of points for me. We re-up'ed for another year.
The Amex platinum is a slightly harder case for me to make because it's not obvious math. Lounge access is superior and the centurion lounge really stand out. I try hard not to assign values to luxuries I'd not pay for but it's nice! Anyway, back to numbers. The amex offers feature should not be ignored. This feature let's you opt into offers that provide additional points or cash back (e.g. spend $100 at a supermarket, get 1500 bonus points that I value at $22.50). The amex app tells me I've saved $190 in cash (not counting pts) in the last 15mos I've had it so I've made up the net fee right there easily and have re-up'ed (even while having a Citi Prestige and CSR and an amex business platinum but that's a product of our extreme travel hacking).
I occasionally book airfare on it because 5 amex pts per dollar is fantastic but otherwise don't put spend on it. For what it's worth, Citi prestige gets airfare for me despite fewer point because the travel insurance is so superior. For intl car rentals, if you're in a country not covered by CSR or don't have it, the amex opt-in $75k primary opt-in car insurance for $19.95 / rental is good to have. Having a relative whose credit card didn't cover a $15k stolen rental made the complexities of into car rental obvious to me.
Somewhere there was an thread about whether people have AAA. We drive old cars and skip it, instead relying on the services from amex plat. They've worked great and are superior among the premium travel cards. That said, I've had AAA before for the occasional hotel discount but never used it for the obvious services!
Back to points. The platinum doesn't have good earning and isn't required the same way the csr/csp are with chase but amex mr are very be useful. In the us, that's mostly for Delta but they are also very valuable on some of the more obscure partners. Definitely not as easy a case to make as chase partners. Anyway, the platinum card is not about point earning directly or indirectly the way csr is but don't write off amex points as a valuable currency.
I've not revisited the point math fully in a while because I'm confident I can justify both but there's a case in the back of my mind that lurks... Should I dump all of it and just solely use my BOA travel rewards with no AF and earn 2.625% all the time and have no travel benefits but cease all this travel hacking thinking/obsessing? It's tempting sometimes... But I've been doing this for 8 years or more so in many ways, the skills are developed and it's just a matter of staying on top of the latest developments for me.
