I'm going to try to avoid making this a long post.
Inpatient services exist. There is a huge matter of logistics and payment.
Realistically, to succeed, an inpatient service needs to be a regional referral center. Other hospitals have to be willing to send you their paying patients. That means you have to be willing to take everyone, regardless of how poisoned and how many days they have left on their DRG. That also means, that you have to either be "there" or have someone there who can admit the patient. That doesn't necessitate 24-7 in-house coverage, but it means that someone has to be ready to go and see patients 24 hours a day.
For a place like Pinnacle, that work has been done. Ward Donovan busted his ass for several years, by himself, to get his unit. He now has 3 toxicologists, which necessitates Q3 coverage. He also has a resident staff of 4 residents, which is what allows them the inhouse coverage. I don't believe they have any fellows yet, but they are looking for some.
Also keep in mind what a primary admitting service means. All those "fever" and "pain" and "the family wants to talk to you again" and discharges are all up to you. That seems fine in theory, but until your service is set up and running with a good staff, it has to be done while supporting yourself with other work.
Inpatient Tox pays very poorly. Even with critical care codes the pay is pretty bad. Also, hospital compliance people don't like seeing non-intensivists using inpatient critical care codes. It is totally legal, but requires a fair amount of paperwork. However, again, Ward Donovan and few others have fought this fight, so it is possible. Many of the patient are uninsured or underinsured. That can be made up with good outpatient billing. The outpatient billing is actually pretty good. However, you see lots of "NUTS" (no underlying toxicologic syndrome).
Credentialling committees will give EM/Tox trained people admission privileges. ABMS defines toxicology as an inpatient all-ages specialty. You can admit adults and kids. While Med/Peds would give you more inpatient experience, it won't necessarily help much. Inpatient tox treats tox. You don't treat asthma or other crap that the person also has. The goal is in and out. If they need more, they go to another service. To some degree, the EM mentality of in and out works very well with tox patients. Also, ABIM has pulled out of the tox board.
Last, but not least, most people going into tox are EM trained. Thus the fellowships are built around that. Moonlighting is already set up and academic appointments exist for EM. While it can be done for other specialties, it is tougher. A non-EM trained person may need to bring their own source of funding or may need to rely on a PGY-4/5 salary.
More realistic is to start with an occasional outpatient clinic. Develop a reputation and build. Outpatient pays better and allows better time control. Once billings get established, there is available leverage on the Chair and the hospital to give more man power.
So, it is possible. It takes a serious time commitment and significant dedication to the effort. Getting the necessary salary support until billings are established are the first major hurtle, but can be done. It also takes a person with a good sense of business, unless you can work for an established unit.