No they don't

IMO

I can't see Ottawa wanting to swap firsts for that.

I see you're from the future, you were only from one of the possible futures though... Haha just joking. But I'll just leave my perspective why this is an interesting possibility.
Jets could Trade Beaulieu(UFA) and a nice package to get Joshua Brown(UFA) and get 500k retained, just to get over $750,000 and enough room for a 23 man roster for the Playoffs using the maximum Cap. I would even add a 2nd but it would have to wait till 2024 cause Jets don't have a 2022 or 23 2nd. But other than that it would be too much for a rental and to move up in the 1st round(which would be nice to try negotiate.)
Jets have also been using the 11-7 roster amazingly and I believe they could learn to use this system as a very effective alternate lineup. It would be nice to have 2 Joshua's with the last name initials MB as a pairing. To trade for Joshua Brown as a big 6'5 217 lbs Right handed RD, Jets could add strength and physicality to the Right side. This would put Stanley as the 7th man on the 3rd pairing, but that's why it's interesting. Stanley could still use some development, spread the minutes a bit low but try give Stanley at least close to 10 min. You could still give Morrissey and Pionk 22 min. Give Dillon 17 min and then Brown, DeMelo and Schmidt 15 min. What's also interesting is Schmidt plays both sides so we could give him Shifts on both sides during the game. Schmidt could still play some shifts with Morrissey(who he's been playing amazing with) and he could help Morrissey on the left side with the Transitional game and Skating Ability produced to help with plays. Dillon and Stanley are average in those departments but keep a much needed Physicality on the Left side.
Even if Jets are just getting Joshua Brown for a rental, it would be very good for a strong playoff run. Sens don't look like they are making the playoffs this year with a very competitive East. If Brown intends on staying with the Sens future and Sens already have interest in signing Brown big money long-term. It won't hurt the team with a lot of cap space to work with, and Brown could still get paid by the team he's been growing with. It wouldn't hurt to let him have a run at the Cup, possibly win and get paid good long-term by the team he wants to stay with. That playoff experience would be beneficial for the player and his team. Which Sens don't look like they'll be getting that valuable playoff experience, again. This is why I believe this is beneficial for both teams, rather than wasting another year.

Joshua Brown isn't worth that much as a rental. So It really isn't too much value to have to give up. I would've added a 2nd but it would have to be a 2024 2nd because Jets don't have a 2023-24 2nd. Also Simon Lundmark is a 21yr old 2nd Rounder(51st) RD, and is a decently big guy at 6'2 201 pounds. Lundmark is prospect ready and you don't have to wait 2-4 years for him to Develop, and you could still have him waivers exempt for 3 years. Beaulieu is from Ontario and would probably stay with Sens at a low price if given more responsibility and minutes. Beaulieu could also play both sides. Along with 2 3rd round picks which are valuable, saying other wise is ignorant, so adding a Jets 2022 1st round pick to swap with Sens 1st isn't too much to ask for, it's still giving a lot. But other wise it's stupid to pay way too much for a rental.

You kind of summed it up in your last sentence. It's a pointless trade for Ottawa and an even more pointless one for Winnipeg.

Neither team would consider this