"Conditioner" is not for any wood, it's intended for woods that are problematic when it comes to staining1. If you're not staining they're not a relevant part of a finishing regime, ditto if you aren't using a problem wood. Oak is not one of those woods.
If you haven't removed all the previous finish the wood is not ready for staining in the conventional sense so either you're already planning on using something like "gel stain" or you need to switch your plan to it, or a product that works similarly2. Or, you need to strip the piece to get down to bare wood so that you can stain it conventionally.
Note that staining is not a must-do. Stain is only to change the colour of wood if desired.
It is unclear to me as to whether wood conditioner is only for totally unfinished wood
Yes it's only for bare wood. I would hope this would be specified clearly on the on-tin instructions, although I wouldn't be surprised if this is not the case.
The reason it's only for bare wood is the same thing can be accomplished using any finish, including a layer of oil, dilute shellac or thinned varnish3, at much lower cost.
1Woods prone to blotching when stained, e.g. pine and cherry.
2Doesn't colour the wood directly but adds it as a surface coating.
3This is in fact what most "wood conditioners" are, nothing more or less than very dilute shellac or varnish of some kind.