There can be a bit of uncertainty when I brew finishes a few points higher than expected. My last few have all stopped at 1.014 where really, based on the attenuation specs for the yeast used that should have gone a few points lower. E.g. a very recent pilsner fermented with Mangrove Jacks M54 yeast which is supposed to attenuate 77-82% didn't even achieve 70% attenuation. Hard to know why given the fermentation itself seemed healthy enough. I've had onbe 'properly' stuck fermentation that stopped at 1.021. Nothing I did could get it going again. One thing I wasn't in a position to try but is one thing I've read that does actually work is as you've suggested... the yeast cake, or perhaps are jar of the slurry from another completed batch.
Partial mash's are a great next step and in fact they are what I do 90% of the time. I usually use a 60/40 ratio of grain/extract - so in my case I'm mashing around 2.4kg grain typically. The main advantage is that you can experience the mashing process, get familiar with the wide range of grains etc etc and meanwhile keep everything at a stove top level - all you need is a large stock pot and perhaps a grain bag. To me the end result of a partial is indistinguishable from a full AG brew once your grain portion is up to a certain level. In your case you could in the first instance continue with your extract kits but simply replace the brew enhancer/ brewing sugar additions with a suitable base malt instead. For example, with a 10 litre pot you could mash up to around 2kg of a pale malt grain which will give you around litres of fresh wort. If you then sparge with a few jugs full of hot water you could end up with around 10 litres(ish) which you can then and combine with the contents of your beer kit can and top up to the usual 23 litres.
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