This kind of problem is so frustrating to deal with. Sadly having removed the phone line you've lost the one tool I have always found handy and that is being able to hear noise on the voice line.
Noise can of course be on the line or in your mains supply. Running on an extension lead off a spur with an old freezer on the same circuit.... you get the drift.
Trouble is we don't know if it is your modem dropping or the line dropping. You don't say what router you are using inboard of the modem.
Quick and easy changes - you say you've done the cabling bit, so mains supply is worth looking at as well as proximity to cabling in the wall, other electrical activity in the house and so on.
I use DSL Stats to monitor and log line activity if things go wrong. It helps when looking for patterns and clues.
Next, phone socket - has it been replaced recently? Remove the faceplate and direct off the socket.
How does the line come in to the building? Cable trapped, kinked, under water? Mine is connected via a box on the electricity pole in our garden - it's a 50 year old supply! Gets wet, corroded and generally nasty.
I had loads of issues and it took a number of engineer visits all of whom found various connection boxes under the roads/pavements that were full of water.
How far are you from the cab? Any recent building work, new houses or other disturbance.
Yes I did get the 'No fault' £150 threat, but armed with logs, list of things I had done to try to fix and the right approach to the engineer we got there.
The final item was one from a couple of years back when a house was having a new line and the crossed it over with mine in the cab. Didn't re-seat my connection properly and so it kept dropping.....