Success! All switches in and working :)
The flow I eventually fell into was to place a switch in lightly and press it to test it made contact (by seeing the key appear in the text editor) then press it in without worrying whether I got the pins in right.
I haven’t gotten to the Lego case yet but this is currently a fully-functional keyboard. (Albeit not the most comfortable to write on.)
Time for some dinner, me thinks. Will continue later :)