One of the biggest benefits of going with all ductless mini-splits is that every room can be controlled independently. If you aren't using a room, you can shut off the climate control in that room. This can translate to major energy savings, depending on how you use the rooms in your house. Additionally, in many houses the ductwork snakes through the house in ways that cause significantly restricted airflow, which is not a concern with ductless units.
On the other side of the coin, one of the biggest downsides of going with all ductless mini-splits is that every room must be controlled independently. It means you need to keep aware of what the rooms are set to, and remember to change all of them when you need them changed. Different systems have different rules they follow when some units are calling for heat while others are calling for cooling. Some units, such as those made by Daikin, can have a WiFi module installed (in each indoor unit) that will allow you to manage all the rooms from an app in addition to the remote control, allowing you to change all of them at once. Others, such as Mitsubishi, have other network capabilities, but I don't know the details.
In terms of installation costs, those depend very much on your local market. You'll have to get quotes from local installers. You don't need to remove the ductwork -- that will almost certainly be very expensive, just seal it off and leave it there. The simplest seal (on the room end) would be a magnetic sheet like this, but that doesn't look so nice and it only works if your vent louvres are pretty flat. A nicer option would be to remove the louvers and patch the wall. Either way, you'll probably want to put some kind of fire-blocking inside the duct so that fire can't spread easily from room to room. You'll also want to make sure you seal up the furnace (trunk) end of the duct, which you can probably do easily with the same sheet metal the duct is made from.
One other possible downside of a ductless system is that it is typically cost-prohibitive to install an indoor unit in hallways or bathrooms, necessitating that those rooms are heated/cooled by other means (or by spillover from the conditioned rooms). One way around this is by breaking up the existing ducts into smaller groups and having some of your indoor units be ducted instead of ductless so they can handle more than one room. However, that's going to be a lot more expensive to set up.
Despite the potential downsides, I would be inclined to lean toward the mini-splits in your situation.