The stoppage is not caused by vent issues, but by clogged drain lines, probably in multiple locations. The comprehensive solution is to clear all the lines starting from the traps and ending at the street or septic tank. You may get by with less, but dislodging one blockage can move it to another partial blockage and plug the hole completely. You could scope everything first with a camera, but that may cost as much as running a snake through them without accomplishing anything.
I do not understand the assertion that the stoppage is caused by a vent problem. The purpose of a stack vent is to prevent exiting wastewater from emptying the traps by siphoning all the water out. A blocked vent, or the absence of one, does not cause drains not to flow. Wastewater will fall and flow downward due to gravity, and either push along or go around any air in the pipe. That you were able to restore flow by poking the blockage - which you could actually feel - speaks to this principle.
That the stoppage occurs at multiple locations indicates either a block on a shared drain line or multiple blocks. In a system where waste has built up over time, neither would be unexpected. There are many common causes, including worn out garbage disposals sending large chunks or fibrous food waste down the drain (or gummy foods like pasta, or cooking grease, especially when it reaches cold pipes and congeals); insufficient toilet flush water water; flushing kitty litter, foreign objects, paper towels, or hygiene products; corroded cast iron pipes creating rough interiors that catch debris; and insufficient slopes to keep the water moving along.
On the way to the street or septic system, the main line can crack or separate (especially with clay pipe) and allow roots to infiltrate where they form a mesh strainer that catches debris, eventually blocking the flow.
A word of advice if you use a drain clearing service: Expect that unless otherwise negotiated, they will only poke a hole in the blockage large enough to restore flow for the time being. That all but guarantees a return of the problem in time. Expect an offer to scope the line with a video camera, sometimes discounted or free if you have additional work done; it will most likely show root infiltration in breaks and cracks and you will be advised to replace the line for over $10K. I no longer root for the franchised drain service that I once employed (no pun intended).
In my experience, you can avoid that by simple use of a foaming root killer immediately after auguring the line, followed up by annual reapplication at the start of each Spring. (The roots are most susceptible when they are damaged or growing new tips. The manufacturer says the herbicide is not systemic and thus will not harm the tree.)
(If you decide to snake a toilet in the future, you might want to consider buying a closet auger because it works without removing the toilet.)
