I've tried: OPA2134, OPA2227, OPA209, JRC5532, JRC2???, some TLV6xxxx, AD8610 and AD8620, OPA627 and last but most important, ADA4898.
OPA2227 is a good FET opamp. Its main strength is ease of implementation - it's very hard to mess up as it is not fast - and extremely low current noise, so you can use it with tiny resistors to reduce thermal noise. It can run out of small supply voltages too and is not too current hungry.
It's main faults are so-so voltage noise performance, PSRR, bit much overshoot, low output current at higher supply voltages and that it is whole 2V from power rails.

Opamps have very little of a signature really in comparison to the rest of the amplifier unless the circuit is very unsuitable. The main difference is whether it's current feedback (uncommon, e.g. that JRC2???) or voltage feedback. These are not interchangeable. The other important part is whether the opamp is unity gain stable. Many used in audio are.

Then there's the choice of input stage, which is the most important factor in THD. JFET, MOSFET, BJT have different sound signatures - harmonics ratios, with BJT sounding the most analog-like with decreasing harmonics, JFET having "flat" harmonic distortion - lower but higher orders, MOSFET having more odd order distortion.
And of course whether the opamp is biased into class A (more noise, high power usage but lowest distortion) or ran push-pull (with less even order, but maybe more odd order distortion). Extra feedback can also help reduce distortion at the cost of more resistor noise (important at high sensitivities/low impedances).

The differences can also be heard in ringing/overshoot (usually smeared or etched detailing), noise floor - blackness, microdetailing; and crossfeed - both soundstage width, depth and less distortion (crosschannel IMD) giving better detailing. Sometimes subbass if you're just "blind rolling" as in without checking or adjusting feedback loop, resistors, supply voltage and current etc. as the filtering may change slightly.

It is important to voltage match as precisely as possible, otherwise there is a bias towards the louder sound.

AD797 is a finicky beast - it's easy to go wrong with it. It's not unity gain stable, so to use it at low gain you need to add an input cap and slightly damage the performance - even good caps can have some sound. It does sound great when properly implemented though.

I've actually listened to it vs ADA4898 two days ago. Both are technically great, but the latter is slightly better (!!!), like just the next version of the former and way easier to implement. Unless you really need to work on +/- 20-something V which it doesn't support. In fact, it works perfectly well in a textbook design with only a minor mod - removal of ground plane from below the opamp to reduce parasitic inductance. (Putting it in an adapter usually does the same.)

There are ways to make a discrete amp on par with an opamp (hey, it will probably be one made larger), but definitely not better unless you need huge power handling. Some people prefer tubes too, I rather like BJT sound more than the extra 2nd order harmonics tube/transformer sound - but of course there are expensive tube/transformer designs without that drawback. And they sound great, but I'm not rich enough to just go about and grab an amp worth a decent car right now.

Implementation of most opamps is trivial, just keep element values low to have low resistance and thus low thermal noise and parasitic capacitance/inductance, but high enough to not have excess ringing. Or instead, one can add an optional small capacitor in a feedback loop if it's voltage feedback and has slew rate to spare. (most are and have) The main problem with many opamps is low current capability - so you might need to buffer them and there are scant few really good buffers that will not reduce the performance.

AD797 is actually decent in this regard, at lower supply voltages it can source plenty current, but it falls off a lot at higher voltage - but it's typically not required then. ADA4898 can source 38-40 mA always, has a great internal current mirror buffer.

On the rest of implementation, there are some tricks like guard branches (aka Kelvin contacts) if you can spare the inductance.

Hm, the DC offset is unlikely to be high given any modern opamp either way, even if the lousy buffer amplifies it a lot.
Buffering with plain old push-pull MOSFETs is pretty dumb - waste of opamp's capabilities. Even the old BUF634 is better at it, as in less noisy. (As in Pimeta, FiiO E10 or the implementation failure of a Pimeta clone that is called AMB Mini 3.)

Headphones aren't speakers, so you're not in need of large output current at all - even something insensitive like orthodynamics would need a score of mW power.
While AD8597 has very low output current capabilities, it does support more than enough for most sensitive headphones, so buffering it is not really necessary as long as high enough supply voltage is used. That is, unless you want to drive orthodynamics, the least sensitive low impedance headphones out there. It's a bit outdated, but pretty good actually except the output current. It can use slightly more compensation than the datasheet suggests to not ring at small signals.

See, to drive the worst case Hifiman HE-6 to 115 dB (a reasonable level for a hard impact), you need about 2 W power, therefore ~130 mA at 15V (+/-7.5V) (2W) per channel. Almost all opamps cannot do this unbuffered. It takes just about every ounce of power out of BUF634. Of course you'll also need near zero output impedance too.
But to drive something easier like HE-500, you only need ~30 mA at 15V (600 mW), which quite a few opamps can manage without a dedicated buffer, but not that AD8597, which can only source ~18 mA. Some opamps can source more current at lower voltages, but it is not one of them.

For comparison purposes, Leckerton UHA-6s mkII with OPA209 can source ~300 mW (+/-6.3V), which is enough to drive HE-500 to ~110 dB peak. With ADA4898-1, it could do the requisite 500 mW for 115 dB peak if not for the regulator, so it tops out at 113 dB. Note that this isn't RMS power as stated on the site, only peak power. You can do the calculation in RMS too and it works just as well - ~60 mW RMS @ 38 Ohm with OPA209, 105 mW with ADA4898-1, but this hides the fact that the power regulator will current clip on some peaks.
No, it doesn't get any louder, because output voltage stays the same - but it is less liable to current clip.

The grainy/etched/artificial sound is an implementation mistake - typically too high parasitic inductance or not enough compensation (capacitance parallel to feedback) causing large overshoot and/or ringing. Most opamps can be compensated enough to not do that - unless they're too slow/have too low gain bandwidth product. Alternatively, if you don't care about it, adding output resistance will stabilize such issues, but will reduce output power and add more noise. That's the cheap chinese way out of the problem, unfortunately popular.

Disclaimer: back of the envelope calculations. I wasn't precise.