You’re pretty much correct but not completely.
LSE start assessing applications pretty much the moment they receive them, and they don’t wait till the equal consideration deadline. It’s why every year they send out a huge batch of rejections almost immediately in October/November, and have done so this year too with quite a few applicants already having been rejected since mid October (due to qualities of their personal statement). So obviously they have started assessing applications and also sending out rejections, but they only start giving offers till later on. According to past years spreadsheets, the first few offers come in around mid December to early January (before the equal consideration deadline) - but the bulk of the offers do tend to come out between February and May.
Therefore an applicant who’s applied in September won’t get an offer till January at least, but still might get rejected at this point in time - so I wouldn’t worry about receiving an offer/decision anytime soon, unless they deem your PS to be a definite no no and reject you immediately.
Also by law, all universities (inc. LSE) have to assess all applicants who have applied before January 31st with equal consideration. This doesn’t mean waiting until the deadline to start assessing, but instead universities have data to calculate how many offers/rejections they can give before certain date, and how many more applications they expect to receive.