1. I do not believe he wanted to be caught. People of exceptional intelligence often believe they can pull off the perfect crime (Brian Kohberger), and are very indignant when caught. (I am still shocked he planned and pulled off an almost perfect hit, to the not think through "post-op." Beard, Mustache, Hair Dye or even just ditching the mask and wearing a ball cap ...and he likely goes unnoticed.)
2. The best work an attorney will do for him is to get him into a min-sec facility at Butner or Eglin AFB. He has a decent shot at it with his education and lack of prior arrests.
3. It was clear on Day 2 or 3, the NYPD had no clue who he was. I expected him to disappear and his manifesto to be printed in the NYT in the next 6 months or so.