Hopefully the third plumber trip was free, since the second guy did not do the job right. And service charge for second visit should have been free since first one did not do the job right. And you have a lot more patience than me - I would have quit calling that company at least after the second guy.
Assuming the water heater drain was not left open and is draining hot water into the sump continuously, hard to tell where it is coming from. Obviously, if you have a wet spot issue in the yard (presumably near the flex pipe outlet), then the sump pump is "working" - pumping out water.
The discharge from the sump pump has to be to a drain line (illegal in most areas to drain into sewer) emptying to a point where it does not hard - or into a drainage swale or slope leading away from the house (and not toward your nieghbor's house).
Assuming there is no leaking pipe draininig into the sump, sounds like maybe they set the limit switch or pump inlet a bit lower than before, so it is pumping the water level lower than before - AND that evidently is where the groundwater table is in your area, so you are now pumping the groundwater table down with frequent/continuous pumping whereas before the setting was just above the natural water table, which was free-draining out from under the house without pumping except after significant rains. IF that is the case, setting the limit switch a bit higher (but not so high it is up into the motor part of the pump or so high it is above about 2-4 inches below the bottom of your basement slab), you might reduce the pumping a lot.
After pump has emptied the sump, you might see a prior water level line in the sump - where it used to pump out when the water reached that point - setting the switch so it replicates that refill level should put you back where you were before as far as pumping volume, and minimized the risk you will set it too hgih and get a wet basement slab - especially if it is not bare concrete, as that could damage flooring.
I hate to say it, but the sump pump may actually have been working OK originally - if the sump was not filling up with water when he emptied the tank and that caused him to say it was not working - because it may have been draining by gravity through the base material ("gravel") under the concrete slab and foundation - if ideally built for the basement slab to drain to a free surface (common with daylight basements), or if you have a perimeter french drain around the foundation, generally the base material will drain out water under the slab without the sump pump operating, so it operates only if you have heavy rain accumulation around the foundation during rains or heavy snowmelt.
Obviously, I would find another well-rated and reviewed contractor to address the problem any further - bearing in mind the tendency is to sell you their product even if that is perhaps not needed for your case.
If your sump pump is working OK and keeping the water below the slab (ideally at or below the bottom of the underslab pipe outlets - which you might adjust the limit / float switch to achieve), then the issue is getting the discharge to a point where it does not cause you problems - possibly by extending the discharge hose, or possibly by moving it elsewhere to drain, or by replacing it or some of it with a drainage swale leading away from the house.
BTW - the extended discharge line is a BAAAD idea if you ever have freezing weather - because if it freezes up you have a blocked sump pump discharge, and basement flooding. Ideally, the discharge from the pump right outside the house should be into a discharge point which can overflow if the discharge drain pipe outside gets blocked/frozen - with surface drainage from that point to a safe discharge area. This generally means the pump piping to the outside should discharge at a point where the slope is away from the house - not to a "high side" yard.
