While there are some advantages to pagination (some comparisons can be found on this site), I don't think that they are relevant to this case.

A pagination approach that could be relevant is to not just list the pages (1,2,3) but to list the letters (A,B,C). Then there could be a reason for a user to jump to a specific "page". Otherwise there's no point in going to page 24 since you don't know what's there.

This could also be achieved by providing a filtering dropdown with the alphabet at the top of the list. But you already have a search field there - which is a much more efficient solution, assuming that it filters the list on the fly.

Deciding between loading items as the user reaches the bottom of the list, and loading them at the press of a "load more" button is mostly a question of performance. In terms of UX the first option is preferable, because the users don't get anything out of having to press the button. But if the button makes the performance smoother, it could be an acceptable compromise.

While there are some advantages to pagination (some comparisons can be found on this site), I don't think that they are relevant to this case.

A pagination approach that could be relevant is to not just list the pages (1,2,3) but to list the letters (A,B,C). Then there could be a reason for a user to jump to a specific "page". Otherwise there's no point in going to page 24 since you don't know what's there.

This could also be achieved by providing a filtering dropdown with the alphabet at the top of the list. But you already have a search field there - which is a much more efficient solution, assuming that it filters the list on the fly.

Deciding between loading items as the user reaches the bottom of the list, and loading them at the press of a "load more" button is mostly a question of performance. In terms of UX the first option is preferable, because the users don't get anything out of having to press the button. But if the button makes the performance smoother, it could be an acceptable compromise.