Son is full time student w/part time job (earned over $4400 in 2022). Doesn't live with us but we pay half rent and all expenses. Why can't we claim him as dependent?

If he is age 24 in 2022 and had gross income of $4,400 or more then no one can claim him as a dependent.

If he is under age 24, a full time student and does Not provided over one-half of his own support then you should be able to claim him as a dependent under the Qualifying Child rules.  Make sure that you selected that he lived with you for the whole year since he is only temporarily away from in school.

To be a Qualifying Child -

1. The child must be your son, daughter, stepchild, foster child, brother, sister, half brother, half sister, stepbrother, stepsister, or a descendant of any of them.
2. The child must be (a) under age 19 at the end of the year, (b) under age 24 at the end of the year and a full-time student or (c) any age and permanently and totally disabled.
3. The child must have lived with you for more than half of the year. Temporary absences while away at college are considered living with you.
4. The child must not have provided more than half of his or her own support for the year.
5. If the child meets the rules to be a qualifying child of more than one person, you must be the person entitled to claim the child as a qualifying child.
6. The child must be a U.S. citizen or U.S., Canada or Mexico resident for some portion of the year.
7. The child must be younger than you unless disabled.

If he is under age 24, away at school is considered time living at home. Say he lived with you all year. 

If your son is a full-time student under the age of 24 then you need to say he lived with you the whole year.  Living away at school is a "temporary absence."  You can still say he lived with your and claim him as a dependent.

MY DEPENDENT HAD A JOB

If your dependent has a W-2 for his after-school job, summer job, etc. you do not include the information on your own return. You can still claim your child as a dependent on your own return.  He/she can file his own return for a refund of some of his withheld wages (he won’t get back anything for Social Security or Medicare), but MUST indicate on it that he can be claimed as a dependent on someone else’s return.  (Supervise this closely or prepare it for him!)

If your dependent’s earnings were over $400 and were reported on a 1099Misc or 1099NEC then he must file a return and pay self-employment tax for Social Security and Medicare.

You might also want to use free software from the IRS Free File versions:

https://apps.irs.gov/app/freeFile/