Don't worry about memorising every reading for a kanji. They're not all equally common, and if something genuinely does have a lot of kun readings (e.g. 生), chances are that a number of them are limited to specific vocab words. e.g. き、for 生, I think 生地 is the only word I actually know.
In fact, on the joyo list or in some of the supplementary materials, I have seen documents which give some separation of the readings by school year - e.g. a primary school student would be expected to know 生 but not necessarily to be able to read every single word which contains 生, whereas a high school student might be expected to recognise some of the more esoteric uses, and some obscure/non-standard readings would always be given furigana if you're strictly following the joyo list.
To make your life more complicated, I'll point out that some words do not use on/kun readings (in fact, a lot of very common words are in this camp, e.g. 今日). Learn a few common/useful words which use a kanji, and you will naturally pick up the most common readings as well as the exceptions. The ones which aren't common you can leave aside until such time as you need to learn that vocab. In this case, if you're not deep into keigo/business language yet, I'd question whether learning うけたまわる should be a priority.