I didn't go through 141 for my training. I thought that was the only way. If there's another way I'd love to hear about it!
This is a common misconception about the Restricted ATP. 141 only applies for the total time reduction to either 1,000 or 1,250, depending on the number of college credits of FAA-approved coursework; for this reduction to apply the college coursework must have been completed at a college associated with the 141 school. However you can get a Restricted ATP at age 21 and 1,500 hours total time with only 200 hours cross-country time regardless of which part your flight training was under. So you could definitely go to the airlines at age 21.
Since your goal is charter and corporate, applying for SIC positions in single-pilot corporate aircraft is a good idea. Hopefully the company that already accepted your resume works out, but if not definitely apply to others you know of. I have never done that type of work but my understanding is those jobs are quite hard to find unless you know the right person.
If that doesn't work out, you might try applying to 135 operators that hire FOs, such as Air Cargo Carriers
http://www.aircargocarriers.com/careers/ . A friend of mine was recently hired at Air Cargo Carriers with a new commercial certificate so you would have a sporting chance there. 135 passenger carriers like Cape Air, Boutique Air and Southern Airways might be a good option as well. These sorts of companies would be a good option as well, although I know Cape Air requires 500 hours for FOs, not sure about the others. But at these companies you would be flying multi-engine and/or turbine equipment in a crew environment, and in the case of passenger operations you would be interacting with passengers quite a bit, which I would think would be excellent experience for corporate flying.
Another option would be to look into aerial survey. The Pictometry vendors and a few other survey companies hire very low-time pilots, and you would definitely have a chance at the Pictometry vendors with your times. Most of them are listed in this thread, which also includes quite a bit of information about them:
https://forums.jetcareers.com/threa...for-2017-2018-season-fresh-commercial.251351/
Unfortunately they do most of their hiring in September or October as the busy season runs roughly from October to May, so it is probably too late to get hired for the beginning of the season. But it would still be worth applying to these companies anyway, as they will occasionally hire mid-season to replace pilots who leave before the end of the season.
It is also possible you may be able to get hired for a sky-diving or banner-towing operation, although I don't know as much about these operations.
I know you said you do not want to instruct if possible, but if it is financially feasible it would be a good idea to start working toward your CFI certificate while applying for jobs you are qualified for now, in case you can't get hired at any of them. There is a staggering amount of competition for non-CFI low-time jobs so you may not be able to find one, and you will be competing with people who are willing to work for very little or even for free to build time. With a CFI certificate many more opportunities will open up to you, especially in today's hiring climate where most flight schools are hiring and many have raised their pay to attract CFIs. As others have mentioned, instructing is excellent experience as it will require you to have a high degree of knowledge; it is also good CRM experience as you and your student are a sort of 2-person crew.
Best of luck with your job search.