You're in the right here. Both about the car seat needing to be used properly, and about not putting a blanket under the straps.
After making sure that the car seat is the right type for the child's age, weight and size and not damaged, which it sounds like you've done, the most important thing is ensuring that it's used correctly, and snug fitting straps are one of the most crucial things. Unfortunately, loose straps can be a huge problem as they can cause the child to be ejected from the seat in a crash. In this scenario, it doesn't matter how protective the car is. The car can only protect the child if they are properly restrained. Incorrect usage of car seats is implicated in the majority of child road casualties as passengers. The only thing worse is not using one at all.
If your child has only just come out of their infant carrier, then it might be that in this larger stage seat, they are supposed to use it with an insert which reduces the seat down to the size of the child. This is probably the idea that the relative is going for with the blanket - it's well meaning, but unfortunately not appropriate because the proper inserts are very highly tested and designed in order to maintain the appropriate level of restraint for the child, even under crash forces, which can be very violent. You will never get that level of precision with a random blanket that you have put in yourself. One issue with understanding car safety is that most people don't really understand exactly how violent a car crash can be. However, most people are familiar with falling - a good mental exercise is to say imagine that the car was dangling, nose down, off a high surface like a bridge. With the car seat set up as it is, would you be confident that the child would stay in place? Then imagine that the car is being shaken up and down. Again, would they stay in place? The amount of restraint that you'd need in that (imaginary!) situation is not far off the amount of restraint that you're aiming for in a car accident for the outcomes as designed/tested by the seat manufacturer.
Without knowing the exact seat model I can't really say further but perhaps you can get hold of a manual for the seat, these are generally available online, as this should give guidance on any inserts for younger ages, as well as how to adjust it.
The other adjustment which might need to be made is that the straps on the seat might be set up in their position for a 3-4 year old child at the end of the weight range of the seat. Usually they can be set to several positions and should start out in a lower position and be moved up as the child grows. The correct shoulder strap position is roughly level with the child's shoulders.
What you do find on some second stage seats, especially if they are forward facing, is that even the lowest position of the safety harness is not actually low enough for babies at the lowest end of the weight/size range of the seat. In this situation, you need to keep them in the infant seat or look at a combination (e.g. birth to 4 or Group 0+/1 seat) which covers both stages, and probably keep them in the "younger" mode for now if they fit the requirements, which will be more protective.
Lastly I won't go on about it, but I would point out that if your child has just come out of an infant carrier, then they really ought to be rear facing - it's much much safer, and babies at that stage, around 1 year old, are very vulnerable to injury in the neck and spinal cord, which they are at risk of in an accident in a forward facing seat, even if it's used perfectly.
If your baby is under 15 months or under 9kg, then they MUST be rear facing, by law. Over this, it is your choice as the parent, but rear facing is still very strongly advised.
If they are forward facing, then it is even more crucial to ensure that the seat is set up correctly for them, which means:
- Ensuring that the seat is fitted correctly to the car: all isofix points green, leg extended or top tether used, or if it's belt fitted, correct belt routing and sufficient belt tension, which might need an adult to kneel into the seat to fit it.
- That the safety straps (and any relevant inserts) are positioned correctly for the child and tightened so that they are a snug fit
- That the child is not wearing excess clothing, and nothing is between them and the straps
Would it be possible to fit your own car seat into the relative's car?