Coffee is fairly oily and IIRC some of those oils are not good for your heart (although caffeine itself isn't bad for it.) You can make up for this with thicker paper filters but tea is a little healthier.
Any paper filter is fine. The lack of one is what will really do you in.
Following the irreparable failure of my poorly designed and manufactured drip coffee machine, I tried the French press and kettle I'd bought for a camping trip years back and, after that trip fell through, never found occasion to use. The coffee was much better, with a creamy taste and mouthfeel that I'd never experienced before in the beverage, so I didn't bother replacing the drip machine. About a month later, I had my first gallstone.
Since then, I've been using a paper filter - the same cheap recycled ones I used in the drip
machine - along with the mesh and plate filters that come with the press. It cost me the creaminess that came with the oils which a paper filter removes, but that first gallstone has also so far been my last, so it's a trade I'm happy to make. Kidney stones are much worse, as I have also had the misfortune to learn, but that doesn't make gallstones pleasant.
That said, even with the paper filter, the coffee's still quite a bit better. Between that, my newfound mistrust of drip machine engineering, and the pleasant physicality that assembling, using, and cleaning the press provides in my morning routine, I expect to stick with it indefinitely.
(You can see the difference between filtered and unfiltered coffee. Unfiltered, it has a deep golden sheen caused by light reflecting from tiny droplets of oil emulsified throughout the liquid. Quite lovely
to look at, and very pleasant to drink - until one or another of your organs packs up under the load. Given how easy it would seem for oils already so dispersed to enter the bloodstream, I suppose I'm lucky it was just my gallbladder complaining, and not my heart...)
Id recommend you get yourself an aeropress. It uses a paper filter, and the coffee tastes magnificent, similar to french press for the dark roasts I love. And it leaves itself open to experimenting with different steep times and brewing techniques. Its quite fun to play around with.
Regular drip-machine filters work fine in a French press, you just sandwich one into the parts stackup at the plunger's business end. An Aeropress sounds like
fun for coffee experimentalists, but all I require is that my two morning cups be delivered with maximum reliability and minimal fuss, so I'm all set with what I have.
Not to discourage your French press setup (as it sounds lovely!) but I've never encountered "maximum reliability and minimal fuss" better than with an Aeropress. If you're looking for something to try be sure to give it a go!
I mean, the French press makes two cups' worth in one go, which by happy coincidence is exactly as much as I permit myself per day. Having to go through the setup and teardown twice, as the Aeropress's one-cup capacity would appear to require, lacks appeal by comparison. But I'm glad to see that so many Aeropress evangelists have found something that seems to really work well for them!
I actually noticed this because I was using an aeropress and thought it'd be more environmentally friendly to get a metal filter. But then my coffee got oilier.