No need for the front derailleur, and the rear one already has one.

I've had them in the past on bikes, and I never ended up using them. I suppose if you had a mid-race wheel change, maybe it could be used for on the fly adjustment of the rear derailleur, but that's gonna be pretty darn rare occurrence, and you could still just take a few seconds to hop off and adjust the barrel on the rd instead. It's also another piece to introduce a little bit of cable drag along the housing.
As for the front der.. If you've set it up properly to begin with (cable is stretched and FD adjusted when first installing), that should really never need to be adjusted.
.02..........

Nope. The only time I would use them is for specific cases, in particular for a frame that has complicated internal, full length housing and you will be swapping out front ends on a semi regular basis. Then you could use them to leave in the cables in the frame.

But otherwise, no. You don't need them.

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Ok thank you. That is good to hear. I really hate the look of inline barrel adjusters on a otherwise clean looking cokcpit.

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2019 T-Rex Tri Series

Not really, no. At least, not once the cable housings have settled in.

One for the RD can be "handy" during the first couple of rides after putting on new cables/housings. But, its not that big of a deal to stop and add a quarter turn once a ride, if you get a "clickity-click-click" on a down-shift, and the shift doesn't happen. After that period, my barrel adjusters remain in the same position for months.

If you work through the gears enough during initial setup, even that can be mostly avoided.

I'm not sure I've EVER touched my FD barrel adjuster. As mentioned, tension the cable, set the limit screws. Done.

If you prestretch the cables and properly seat the cable housing and ferrules, you should never need to touch the barrel adjuster. I literally just fiddle with the derailleur barrel adjuster when I install a cable, then it only gets touched after that for swapping wheels.

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The Shimano FD-5800 front derailleur requires high cable tension and a lot of mechanics use an inline cable adjuster to achieve that.

Most of the time its not needed. I do have one on my road bike for the rear derailleur though. The adjuster on my rear derailleur got cross threaded so without it I would have no quick and easy way to adjust it.

They can be handy for setting up an indexing front shifter, but otherwise not really necessary.

It's easy enough to get ample cable tension on the front derailleur (just temporarily shim it out or wind in the lower limit screw a bit) but the tolerances on an 11-speed Shimano FD is much tighter than the old 10-speed ones.

Last install that I did, I didn't have an inline barrel adjuster and it was a real pain in the ass to get exactly the right cable tension for good shifting (and trimming) and zero chain rub. With a barrel adjuster it would have been a piece of cake, but I agree with everyone else that they are a big ugly and pointless after bedding in.

They are not required, but I myself definitely prefer a barrel adjuster on the RD for the finer index adjustment. I actually am not sure how you'd do this fine index adjustment without one.

The FD doesn't require a barrel adjuster.

I'll also warn that I've been unimpressed with inline barrel adjusters (the ones that you cut out and replace a section of housing with); they seem to do very little in terms of cable tension changes for me.