TBH I have not read Sarah-Ockwell Smith's books except her second child one which honestly I didn't find very helpful - so I can't speak much to the tooth brushing/bath example.
But I think to illustrate that there are a lot of things under the umbrella of gentle parenting, and why it's an unhelpful term to use:
Some people would follow a playful approach and persuade the child into the bath/toothbrushing by making it fun. Making up games or songs, buying a character toothbrush or toothpaste, coloured bubble bath, new bath toys etc.
Positive parenting would say use the play approach, or praise to motivate, or use a sticker chart to build the habit.
Some people would consider the above too coercive and like "tricking" the child.
A reasoning approach might be to explain about tooth bacteria, cavities, toothache and loss of teeth.
Some people would say that is unfair on a child who can't developmentally understand that yet.
I have actually seen people say they would let the child not brush and then when they get a toothache, tell them it is their fault. I cannot fathom what they think is gentle about this, but I've seen it called gentle parenting. Probably because they think it's a natural consequence.
Natural/related consequences would be to explain to the child that they may not eat any sugar as they have chosen not to brush their teeth, with the aim that the child says oh right, better brush my teeth then! (But also OTOH may be seen as a harm reduction technique if the child actually makes the choice not to brush their teeth).
Or the parent may choose a natural/related consequence of pausing/skipping a later, wanted part of the bedtime routine until the child brushes their teeth.
Some people favour autonomy so let the child do their own teeth. Some say you should then do them afterwards in order to ensure they are sufficiently cleaned. Others may provide feedback depending on the age and receptiveness of the child.
The Sears book (attachment parenting) actually says yes make it fun/try praise/try letting them have a go etc, but if necessary hold down the child and do it and be empathetic about them not liking it. Because with their first daughter they let her skip toothbrushing too often, and she got terrible cavities. RIE likely says this too, though their line would be "Thank you for showing me that you needed my help with brushing".
Some say all behaviour is communication, so a refusal of tooth brushing has an underlying cause that the parent should identify and solve the problem. For example, does the child prefer a different toothpaste, a different brush, a different time of day, warm water, a different method altogether. This might be more common when there are known special needs e.g. sensory sensitivities, autism, communication difficulties.
Some say all behaviour is a sign the relationship is ruptured and it's more important to preserve the relationship/trust than to force children to bathe/brush teeth as often as most people would. This might be more common/appropriate in a situation where a child may have suffered trauma or the relationship is damaged or new such as foster care. It is usually accompanied by a suggestion to do lots and lots of relationship/trust building at non toothbrushing times.
Some say skip the battle right now, it can always happen tomorrow (BUT they also hold the belief that in this case, it's important that it DOES happen tomorrow or by some near point). The point of skipping the battle might be to avoid developing a bad association with tooth brushing/baths, or to preserve the parent/child relationship, or to allow time to work out a solution that works for everyone.
Some say skip the battle right now, it can happen when it happens and they have no timescale but they do expect to get to this at some point. Part of Ross Greene's Collaborative Problem Solving approach (plan C) is like this.
A timed pause on the expectation, to remove pressure, is also part of a positive parenting approach, but again comes with an expectation that the issue will be resolved at some specified point in the future. It usually has a set decided length which isn't communicated to the child.
Some would accept a lesser version of teeth cleaning, because at least it is building a positive habit/expectation, but without a structured plan.
Some would try to acclimatise, like first just get the child to hold the brush (praise), then put the dry brush in their mouth for 1 second (praise) then put brush in mouth for longer, then accept some adult movement of brush, then for longer etc, then add a smear of toothpaste, building up slowly to 2 mins brushing with a pea-sized amount. (This is actually a behaviourist technique called shaping, but it's positive and at the child's pace so likely considered gentle).
Some would literally skip it because they are prioritising other things and simply trust the child will get back to it eventually, but don't make any effort to actively push this. (I think this is a problem).
Some may be grateful for the excuse not to push a battle on their child, or be too tired, or disorganised to have a routine, or too burnt out, or simply not know how to. (I don't think this has any redeeming features unfortunately, whereas I can see all of the others being appropriate in some situation even if it's extreme).
I was told by the founder of Visible Child I should let my then 11yo choose not to bathe, and he could decide for himself if being smelly bothered him. I did not think this was good advice. To be fair she does not call her approach gentle parenting (but many, many followers and fans do).
Sorry that was stupidly long. But see why "The gentle parenting approach is...." doesn't work?