The modern way to progress to chin ups normally involves bands and eccentrics, but you can also just do partial reps (starting with the top of the movement) and gradually work on adding depth.

(This also avoids the stress on your elbows that can happen from the bottom of the rep if not controlled well.)

If you're quite far off starting with bodyweight inverted rows and working up in reps will train your arms and elbows in a very similar way from an easier position.

Anything "partial" gets a bad rep from people ego lifting with weights but it can be a legitimate way to work up to calisthenics exercises and saves fiddling around with bands

For progressing to a chin-up or pullup, have you tried negatives? Or rubber band assist? Or, if you are willing to get rings, you can lower them so some of your weight is supported by your feet on the ground (body straight like a plank), changing the height of the rings will increase or decrease effort.

If you haven't tried single leg movements, that's the way to increase load on your legs when you don't have more weight available.

Moving fast is also a good way to increase intensity without increasing weight (you can clean less than you can deadlift, for example). The training effect is slightly different, but probably more useful for the average person tbh.

Chin up progression is eccentrics and band assists. They'll get you to your first chin up faster than lat pulldowns since you're actually doing the movement.