Here's my analysis on that:
How would it be harder to get more people to join if now people could just share a Discord link, then find the forum and go there. Also, your second point happens a lot more without the Discord because most people have to go to a specific site, instead, they can just go to Discord and see that one of their servers has an update, then probably go to that forum again (increasing retention rates possibly). Also, yes... people getting banned and mods not being able to know who they might be can be solved by adding the 10-minute limit to the server, it may be a little weird, but it gives time for them to really look at the rules, get used to the layout, and then be able to easily talk normally. Also, for your final point, yes... some people may just leave the server and go to other ones, but that's also true for this forum just as much, if not more.
Here's how I would setup the Discord:
First, make them wait a bit before they can start posting, then:
Use a leveling system like MEE6's bot, allowing lower leveled people less access to the Discord server, till eventually, they level up enough for access to all of the Discord server (except for Admin rooms of course), making it so that spam accounts will be less common. This will also add some sort of incentive to keep on posting and may help ween out the users who won't post as much. Furthermore, we can host giveaways that are either Discord exclusive or Forum exclusive, shining a light on both platforms to get more users on both.
What do you think about that?