Most mesh systems are proprietary so you would either have to get something from verizon that matched their router or you would have to pretty much completely replace the verizon device and use it basically as a modem.
First note there is no such thing as cat6e and any vendor that tries to claim that is either stupid or they are trying to scam you. The cable designed to run 10gbit is called Cat6a. You do not really need that if your internet is only 175mbps. You can get by with cat5e or maybe cat6 if it is cheap. Key is to get pure copper wire with wire size 22-24. Massive amounts of fake cable being sold that does not meet the specs. Read the fine print and if you can't find these number shop someplace else.
I would look the part numbers up for your current equipment and see if it has 100m or gigabit lan ports.
So called MESH systems are mostly marketing hype. They are basically wireless repeaters some of the very expensive ones have extra dedicated backhaul radios to get past some of the performance issues using a radio for both backhaul and end user.
Although you can use most mesh systems via ethernet cable you are just paying way too much money to get them to stamp MESH on the box. Maybe they could put "new and improved" and that would make it faster
When you run it with ethernet cables you are using the industry standard for wifi installations that have been done since the beginning of wifi. It has always been one single network no matter if you call it mesh or not. The remote radios are functioning as simple AP.
If you think there is some magic roaming function that is more hype. The end device not the network controls roaming. The devices will or won't move between wifi source as long as the SSID match no matter if it is mesh or AP.
It is much cheaper to just buy a inexpensive router (you are only using the radio part) and run it as a AP. You could also buy real AP if you like.
What I would recommend you try first is to use your current equipment. If they do run 100mbps only and that is causing you some problems then you could look at replacing them. In general most application do not even need 100mbps, espeically stuff that runs on a mobile device that has no option to be hardwired with ethenet. Pretty much the only thing that can easily use over 100mbps is a large file download and I guess it depends on how much of that you do. Stuff like web surfing or even 4k netflix does not come anywhere close to 100mbps.