Why should I try EUNA when I have other buffers in the chain?

The buffer in the TU-3 is not a bad place to start, it is very low noise. But it does have some tonal issues to me, I think in fairness it was designed as a tuner and not as an amplifier circuit and it serves it's own design goals very well.
The input impedance and FET switching front end don't sound very natural to my ears, and the output drive can leave a little to be desired. If you have a long cable run, or a big board, it starts to lose level and low-end as you weight it down.
When talking about it, I think it's better to look at it like an audiophile rather than a better/worse. For me, I was very dissatisfied with the solutions available. I worked for a long time as an audio engineer in recording studios, and on the road, and I found so many situations where a good buffer would have solved a lot of problems, but the ones I was using all presented their own. EUNA is more or less my attempt at the unity-gain circuit I wanted that sounded great and gave me great options when driving down the chain. So I started with the power supply, it accepts AC or DC in either polarity, 7.5 to 30 volts. Internally it goes through both switch-mode and linear power supplies to create +/- 15 volts rails which end up giving you about 30 volts of swing. If we're being technical, it's more like 29 volts because the opamp can't swing the whole width of the rails, and the linear supply is trimmed a bit. This means that there is unlikely to be a guitar out there which could clip the input, even an active one. Famous last words, of course, but it would be tough. A guitar with hot pickups can clip the TU-3 circuit, it doesn't sound too bad but if you're ever wondering why they can sound a little cloudy or dull that's part of it.
The output drive can handle down to 600 ohms without a drop in level or change in frequency response, so you can plug it into almost anything. The TU-3 will lose maybe 5 or 6 dB overall and a good bit of low-end when loaded down. It's not normal to drive 600 ohms with a pedal, but that's the lowest common load of classic audio equipment matching transformers, and I wanted to be able to plug right into a Pultec sometimes.
So, in fairness, it's a little overbuilt. You don't need 20dB of headroom and 600 ohms output drive normally. But building a device that can do those things was my goal, and it may explain why some people think it sounds "better" than a conventional buffer.
Also, conventional buffer circuits are extremely simple and typically don't have a pleasant "sound" or "feel" when compared with an amplifier circuit. The technical details of what is a buffer and what is amplifier are kind of murky, because technically they are both just amplifiers. Buffering or making gain is more of a feature of an amplifier circuit than a type. When you see conventional opamp buffer circuits they are usually just showing you the easiest way to accomplish the goal of buffering with the fewest components.
Next step, I worked up an amplifier circuit that sounded good to me but didn't make any gain. The filters were originally to help balance the response of different guitars, maybe beef up a thin bridge or brighten a dark humbucker etc., but most people including myself just use them creatively like an EQ.
Then you move on to layout and parts selection, which have a huge impact on the sound. EUNA has no electrolytics in the signal path, most people agree that electros don't sound great compared to other cap styles but sometimes are simply the only choice available. This is definitely a thing people argue about, in the audiophile and studio gear worlds, a lot. Some people will tell you all caps sound the same, but that's simply not true. Different caps have different electrical properties which may affect how the circuit works, and some of the more complex interactions between components are really hard to describe without getting into material science. Suffice to say, I designed around the parts I thought sounded best.
Some of the benefits stop at the next active stage, so it will push through all the true bypass stuff but the next active stage is a potential bottleneck. There's only so much you can do about this, and my assumption is always that you've chosen the stages in your board to be the way you want them to be.
In general the audiophile approach is helpful if you can avoid going off the deep end. Look for the bottlenecks in the signal path and decide if they are worth investing in.
To me the front end unity-drive thing was a place in my chain I wanted to invest in, if only so I knew it was bulletproof. I think EUNA does sound better than conventional buffers, but how much better is impossible to quantify. To some people it may be zero! But I am happy with it because I never ever wonder if it sounds good or not - which I did all the time with the TU-3. I would pull it out, measure it, poke it, put it on a switcher, I just never trusted that circuit or liked that pedal. I felt the same thing with other buffers. It may be all in my mind but when trying to stay in a creative headspace I think it's important to build a rig you can trust. That will mean different things to different people of course, but EUNA gives me that feeling of trust.