You can judge for yourself when WK is being annoying and marks you wrong for a potentially correct answer by just using rfindley’s double-check script. Aside from that you can use reorder ultimate 2.0 to get most of the reordering power you seem to want (I’m not sure if it can sort by kun and on, though).
I think one thing you can consider is that a lot of the items at lower levels have mixed readings because both on and kun are common, such as 人 or 口. At higher levels this is not really a thing anymore, as you start to encounter kanji which only have on readings or whose kun readings are only variations of verbs with okurigana. As such, the proportion of kanji you learn that have their primary readings set by the WK team as kun will decrease as on readings become more common.
I had a similar grievance at lower levels, because prior to starting WK I knew about 150 kanji with a jumble of on and kun readings. This eventually ceased to be a problem when the aforementioned drop-off in frequency and kun readings meant that in the levels after about 10 or so you can easily expect to find that almost all of the kanji on a level are given the on reading by the WK team.
(I’m not sure if this is what your problem is, but a lot of people seem to have trouble with it, so I’m leaving the following part in.)
One trick you can do is to try to grasp the differences in the phonologies of native Japanese words 和語 and Chinese origin words 漢語. Typically you will find that, since most on readings were imported from China during the MC period, the syllable structure and general “sound” will reflect the Japanese perception of those loanwords at the time. You eventually come to grasp the patterns: りゅう, きゅう, しゅう, and those “little ゃゅょ sounds” are very common in 漢語, but not necessarily in 和語*. Almost all (afaik) on readings are only one or two syllables long, thanks to the phonology of MC. Other examples are こう, とう, よう. Presumably the double syllables came from when the original MC syllable ended in a type of consonant, as in しゃく (尺 series) or はく (白 series), but this is speculation.
Conversely native Japanese nouns have pretty open syllables, at least from what I’ve seen. 心, 霜, 雨, 鰐蟹… similarly 和語 compounds will have distinct phonological patterns and shifts. For example, 言葉 is not ことは but ことば, because of rendaku, which happens quite often with 和語 but more rarely with 漢語.
*do correct me if I’m wrong. Not a linguist.