To add some clarity on this, If the villager has no profession and is converted it will be unable to get in minecarts and boats. If it has a profession (either from spawning with one or linking) it will pop out but can still get in them. If you place a block above or to the side they will reenter the cart immediately; but they need a profession or they can't enter carts. Trading with them or having an actual workbench doesn't matter, they just need a job. Nor does multiplayer, you can recreate this if you follow all the steps.
RespectKnuckles
To add some clarity on this, If the villager has no profession and is converted it will be unable to get in minecarts and boats. If it has a profession (either from spawning with one or linking) it will pop out but can still get in them. If you place a block above or to the side they will reenter the cart immediately; but they need a profession or they can't enter carts. Trading with them or having an actual workbench doesn't matter, they just need a job. Nor does multiplayer, you can recreate this if you follow all the steps.