I disagree with this. The big pitch from D3 schools is that you are not committed in any way (unless the school has ED). After my daughter signed her D2 NLI, many D3 schools were still trying to ‘steal’ her and kept telling her “You know you are allowed to go to a D3 and play right away.” In this case there is no honor among thieves.
To the OP’s question ‘do you get anything’ from D3 schools?’ Not really. Some give a letter from the coach so the student has something to sign with the other NLI students at the ceremony (but some just fake sign a piece of paper). At my daughter’s ceremony probably 5-6 were going to D3 schools, one to MIT. A few had those letters from the athletic dept but they were not contracts. An official NLI must contain a ‘Grant-in-aid’ (scholarship). Can a coach hear that you are still recruiting and ‘get mad’ like @Mwfan1921 says? Sure, but the Caltech coach may not have a lot of others to replace you with - it’s a small school with a limited number of talented players admitted. The coach can’t withdraw son’s acceptance to the school.
You are going to find a huge, huge difference in the contact between a coach at Cal and Caltech. It is literally a different league of play. I would take the MIT coach’s lack of interest as just that, a lack of interest or at the very least an indication that he can’t do anything for you with admissions. Could your son still make the team as a walk on? Sure. Is he guaranteed to make the team even if the coach called him every two weeks? No. And definitely not guaranteed any playing time.
Caltech is just different for athletics. It doesn’t have the prestige the other schools do for athletics, but certainly makes up for that with academics. The coaches in all sports have to work harder to attract good players and it sounds like the soccer coach really wants your son on his team.