I don't believe a solid college program would rely on a walk-on to fill their catcher position. If your son is hearing "get admitted on his own" then the Ivy's already have their catcher recruited. Sometimes coaches won't recruit a catcher at all if they are solid in the position and have more pressing needs elsewhere (my son heard this from several schools).
My observations from my son's recruiting journey (2019), which included Ivy's, was that coaches looked for catchers with tools who could hit. The Ivy's were no different than other D1 schools. The coaches he was recruited by mentioned they liked size, athleticism (speed, agility, quickness behind the plate), assertiveness/aggressiveness behind the plate - i.e., fielding bunts, back-picks, etc (game footage was very helpful in this regard), and obviously a strong arm and good catching skills... but I don't think we wouldn't have had those conversations if they didn't like his bat. However, it's difficult for a coach to judge the hit tool from video or a camp setting. They can get some metrics and see if they like the mechanics, but they need to see him hit against good pitching. If they haven't seen your son play this may explain their reluctance to pull the trigger on an offer. My son was recruited solely from WWBA, Jupiter and a few other events (preceded by emailed video to get on the coaches player watch list). I think his performance at these events made the difference. He didn't do any showcases.