Yes, many private practice docs have fee for service arrangements and are able to have the type of practice you described. However, building a large enough patient population willing to pay out of pocket and be consistent enough with appts and follow-up to support a full time practice takes time. Just hanging a shingle asking for $X often does not work. Most people cannot afford this setup or want insurance to cover their care. Also, this is going to extremely limit your population given the socioeconomic drift of the mentally ill. If you're wanting to stay interested by your practice, you may need to think beyond this.
Call is often part of a good outpatient system, particularly if you are affiliated with a hospital (salaried or contracted). If so, it is good care to provide extra-hour support by being available to patients and/or other doctors (ie. ER docs). If your practice is associated with any hospital privileges (whether you work on inpt or not) you're likely going to be part of a call pool on some level; particularly to assist in "crisis" issues.
If it was so simple to work 9-5, have no call, charge fee for service and bank $200,000 or more annually, everyone would be doing it. Point is, this is generally not realistically, at least during early career development. Of course, I might not have as much exposure to private practice to be entirely accurate.
