The standard high torque motor is the 37D gearmotor, with 100:1 or more reduction. These are fairly cheap and reliable, and run happily off a standard brushed drive ESC. Joe used them for Grab Crabs v1-3, and Martin Christopher currently uses one to great effect in Toucan, his grab and lift (No link for this, sorry).
Your other options are, like you said, a big servo or a Strangbox.
A big servo is going to cost you a fair chunk of money, as you would probably want 100kgcm or so torque. It’d give you similar performance to a 37D, but with a couple of advantages in that it is significantly lighter, and it has positional control so you can tell it to stop at an exact angle. Jack Franklin won(!) last BBB Champs using one in Luchador (The chronicles of Luchador 1 through 3 - #3 by Jack).
The main advantage of a Strangbox over a 37D is power - you get a grab and throw rather than grab and lift. You can’t get them off the shelf, so you’d have to build one. The process is roughly the same as doing a Rotalink conversion (BRUSHLESS22MM.md · master · Alex Mordue / Saw Loser · GitLab), but you have to do more hunting around for suitable gearboxes and motors, you have to make your own shaft and press the pinion onto it, and theres more of a chance of it self destructing. It’s not an easy task.
For your first bot I’d recommend sticking with a 37D, as it’s the safest bet in case anything goes wrong. Good luck, and have fun building!