There are pros and cons for support considerations and whatnot, but this isn't what I meant by implementation. If you want to do SMS-based OTP, you need to interact with a 3rd-party service to deliver those SMS messages, but all the rest is basically the same, you need to generate key material, produce OTPs and so on just like you would with TOTP. Since there's a 3rd-party integration involved though, I'm considering it more complicated. But I guess you could argue that having to produce your own UI to manage TOTP and recovery codes and the like swings the balance.
That is right, you need to use a service provider for sending SMS. This was more difficult in the past but now there are non-operator providers, Twilio etc., which makes the integration (both technical and non-technical) simpler. Sometimes the mobile operators have a different service to be used by banks for OTPs due to security, availability and performance concerns, this may make it a bit more complicated.
The key material issue is not the same for both. With SMS OTP, you can just randomly generate a code, keep it and its timestamp in the database, and then check when the user enters it. With TOTP, there is a need to share a symmetric key material because TOTP is generated with a HMAC algorithm, so both generator/user and verifier/service needs to know the secret. Sharing the secret can be as simple as generating/entering some numbers/codes, a QR code, or an online key exchange algorithm. When there is a key material, that has to be kept securely, for example banks may want to use HSM. The same is true for the client side implementation, the key material has to be kept on the device (phone) securely, so the imlementation is not as simple as just keeping any data (however it became simpler in recent years).
A service/bank etc. probably already have some SMS service provider integration, so it is pretty simple for them the code this. It probably takes more time to create user documentation for it. TOTP or any such implementation (FIDO etc.) is more complicated, because some things provided by the SIM card and GSM network has to be replicated. Potentially an HSM is involved. Because this is considerable more complex than SMS OTP, the security review will probably take also a longer time.
I might be a bit biased, I started worked on these when TOTP/HOTP specs. were still drafts and OTP/2FA was not mainstream, and worked until TOTP/HOTP (there is also OCRA for signing) became mainstream and FIDO appeared. Now, I guess everybody knows more about these, so the implementatin process might be more seamless than I think.