Conceptually, I really like JWT as it is in line with the statelessness of REST etc (no state saved server-side, all relevant data is contained in the token).
What I am
There isn't a black and white answer to the given problem of how to implement the remember-me functionality theoretical wise. Lots of theoretical arguments are given on why something should not be done, while at the same time there is no clear answer given to the problem as for how it should practically be done.
Remember-me implicitly comes with the problem that you need a longer time window of token expiration, there is just no way around this. The most secure way is to have the user login regularly given brief expiry; nobody practically likes this though, so a tradeoff is made where theoretical secure perfection is weighed against practical measures.
The way this tradeoff works is by patching the imperfections that come with having a long expiry for a token. But make no mistake about it, you will need a jwt / cookie with long expiry (whether you use two tokens, implement some secondary refreshing mechanism or whatever, at the end you run into the same problem)!
Based on articles I've read on how others did it, this is how it's done.
The way I will implement this is by offering a 3-month expiry on a jwt stored in an httpOnly / secure cookie when the user checks remember-me.
On logout just clear the cookie.
Obviously protect using https / CSRF measures on top.
If you don't agree with it, stop fussing and offer an alternative solution - which there clearly isn't from the infinite number of discussions I've read on it.
If there were a straightforward solution to the problem, there probably wouldn't be so many discussions about it in the first place.