He probably just wanted to stay in Boston and was strategic with his NTC by putting teams that hadn't committed to a #1 goalie yet.
I would look at it this way. If a player has a full NMC, that doesn't mean they aren't open to a trade in the right scenario.
His agent apparently tried to maximize Ullmark's trade protection to allow Ullmark to maintain as much control over the situation as possible. If there is any truth to the speculation that he may have put Ottawa on his no trade list on July 1st, that doesn't mean Ullmark doesn't like Ottawa or doesn't want to be here. It means his agent is doing his job by putting Ullmark in the driver's seat.
It could mean that he preferred to stay in Boston and play out the season there, even if the prospect of an extension was low. Or it could mean he preferred to come to Ottawa, but only with a massive extension that he would have the leverage to ask for if he could block a trade. Now if he signs an extension, he has less leverage because he is already here.
So I don't think the take away should be that he is upset about being traded to Ottawa, or doesn't want to be here. If there is truth to it, the take away should be that his agent may have done his job and given Ullmark near-complete control over the situation.
The other factor is that players don't like when trade talks are dragged on. If there was any direct or indirect communication that Ottawa would be on the no-trade list come July 1st, and they were the last team standing, that could have been the agent's way of expediting the process to ensure that the Bruins did not keep Ullmark in limbo until training camp or later. As in, we know you're trying to squeeze the biggest return by waiting Ottawa out, but that's not our problem, he has been shopped since the deadline, either get it done with Ottawa before July 1st or we'll put them on the list. That could have been the context.
A lot of the speculation and rumors are a game of telephone because so many insiders, fake or otherwise, have given us contradicting events. But if we assume the fear about Ullmark putting Ottawa on his no trade list was real, we shouldn't assume it was because he doesn't want to play here since these things are rarely that simple.