I'll start by saying that this is the first time I've heard of the name "Rapley method," but I did a sort of middle ground between that and spoon feeding (basically, I fed my son mashed food, but was simply his hands until he could do it himself, how much he ate was dictated by him, and the process as a whole was baby-led). Here are some of the advantages/disadvantages to the method that I can think of, based on my experience:
- You don't have to sit there and feed baby yourself, allowing you to
supervise while enjoying your own meal.
- Your baby has full control of how much s/he eats at a sitting, fostering trust in his/her own hunger/full signals.
- You don't have to deal with taking the time to mash the food, or spend extra money on buying pre-made mashed food.
- Baby is less likely to choke, because s/he is in full control over what goes in his/her mouth, and where it goes. (Note - I found that my son would occasionally "choke" in the sense that some food would start going down his throat before it's chewed up enough or before he was ready to swallow, but he always coughed it up on his own and soon learned how to not choke and what to do if he does.)
- It encourages following baby's cues for transitioning to solids, which can encourage parents to follow baby for other cues, as well as ensures that the transition to solids is done with the baby is ready, and not by the parents trying to base it on when an article or book says when baby should be doing something. (Many people see the milestones as deadlines, and it can be a cause for anxiety in the parents and added pressure on both the parents and the baby.)
- Things will be messier at first, as baby learns what to do with these new little things you've put in front of him/her.
- You will likely waste more food, because baby has decided to mash it into the table, or various other places, especially before s/he realizes it's edible.
- Baby may not start on solid food until s/he's a bit older. This means breast/bottle feeding for longer, and possibly more frequently, as baby's needs start to outpace what milk can provide, until baby is at least mostly weaned.
- It may delay baby's use of utensils. If you start with finger foods, it may be easy to forget to incorporate foods that require a spoon or fork and doesn't expose him/her to the spoon as early as feeding him/her mashed foods might.
As for your question on whether a child will have enough food and variation - at first, no, most of his/her nutrition will still come from breast/bottle feeding for a while, until you build the variety, and s/he starts seeing the food as actual food (and not as toys that happen to be edible). It's recommended to introduce new food with a couple of days between introductions (regardless of when you start introducing solids, you just may start a little earlier with mashed foods). This means that, at first, baby will only be familiar with a few foods. This will gradually grow in both number and variety as you introduce foods.
That doesn't mean that the "solid food" sessions have to be devoid of either nutrition or taste. I found things like blueberries, avocado, bananas, and steamed carrots to be great starter foods. They're soft enough to not require a full set of teeth, and are packed with nutrition and taste. (On a side note, it's been my experience that the old adage of introducing veggies before fruits so they don't "develop a sweet tooth" is a myth, and introducing and re-introducing the foods is what matters most.)
One thing to keep in mind is protein - most protein-rich foods aren't introduced until quite a bit later, due to their texture. This means that most of baby's protein will be from the breast/bottle for a while. I didn't find it to be a big deal, but if you're finding that your baby is taking to solids and weaning faster than you can introduce protein-rich foods, it might be something to keep in mind, and some ideas for starter foods of this type include tender and thinly-cut/sliced meats (deli turkey is great, just watch the sodium levels, as is pulled pork), cheeses, yogurts (once they're old enough to consume dairy products), etc.
If you've done baby-led feeding with breast or bottle, baby should keep that habit into solid foods, so you shouldn't have to worry about him/her eating enough, as long as s/he can eat when s/he indicates that s/he's hungry and is allowed to stop when s/he indicates that s/he's full. Learning sign language signs for "hungry"/"eat" and "all done" can help with communication.