Alternatives to op amps LM307N and TL071CN
You're right, the LM307N will be very expensive since it hasn't been in production for a loooong time; looking at the schematics top right corner, that's not very surprising: the thing was designed in 1981, and so the TL071 was just out on the market (I think it was invented in 1978).
The fact that there's two different opamps in this device is simply because in 1981, prices for single quantity JFET opamps were probably still pretty prohibitive, and the TL071 of these days probably weren't as low-noise as desirable for an input stage.
The LM307 is a &quot;general purpose opamp&quot; of the early-to-mid 1970s, meaning it's mostly a terrible device, unable to drive significant loads by itself, but also not overly sensibly compensated. That, together with the low-pass behaviour of a reverb coil system, will have lead to the design decision to build a 2-transistor push-pull amplifier with a band-pass feedback structure.
Now, you could simply go ahead and replace everything on the input of the reverb coil with a simple small speaker driver IC (class-AB if anyone asks). You will, however, get a slightly better behaviour (since that will have absolutely no problem driving the input coil) than you get with these ancient transistors - which in turn means that you're not adding another loooooow frequency pole to the reverb coil / input opamp system. Whether or not that makes a difference is not something I can predict – I'd simply try, honestly. +
On the output side: yeah, any audio-range opamp should do if it doesn't have significantly worse noise voltage density than the venerable TL071 (that'd be \$18 \,\frac{\text{nV}}{\sqrt{\text{Hz}}}\$) and a maximum realistic bias current of less-than-relevant (and the TL071 has 200 nA input bias current - you can even get bipolar amps that do that).
Frankly, I'd simply spend 1,50€ on a dual-opamp that's 5V Vcc, rail-to-rail, has lower noise, comparable input impedance, and not worry about having multiple supply rails (+- 15V: where do you even get that?!). The OPA1688 would fit that description. You can simply replace your complete input stage with a non-inverting opamp configuration around that (when you design a PCB, foresee &quot;spare pads&quot; for things like dampening capacitors in the feedback, and add a series 0 Ω resistor on your output - if it works &quot;too well&quot;, you can replace that with higher resistance to make the coil feed back into the opamp).
Output should probably really just be another non-inverting amplifier.
Hint: Replace the ground on the &quot;low side&quot; of your input and output coil with a ca. VCC/2 &quot;virtual ground&quot; (you can use another opamp voltage follower to provide that, but I'd simply use an LDO to get rid of input noise). Then, your whole circuitry can work perfectly fine with a single-sided supply (you're AC-coupling your input, and there's little reason to not AC-couple your output, too), which makes the system as a whole much easier – find any thrown away say 6 V to 12 V power supply you've got lying around, add a bit of input filtering (don't overdo it, modern Opamps have nice supply noise rejection), and be done. No strange +- 15V supplies.

TL;DR: You don't take such a simple 1970's technology circuit and just replace the unobtainable opamps from that. You simply replace the whole circuit with a simpler one.