Eliminating welfare wouldn't increase the number of people desiring to work because most people on welfare already work.
Thus, your argument that eliminating welfare would result in more people seeking jobs, largely fails.
But let's just pretend that you are correct, so that's exactly the reason that it is important that we continue to have a minimum wage.
"A democracy cannot exist as a permanent form of government. It can only exist until the majority discovers it can vote itself largess out of the public treasury." Attributed to Alexander Tytler
As to wage increases, used to be as productivity increased so did wages & that is no longer the situation. Has not been for decades.
Higher paying jobs, one issue are many are without skills or have very basic skill sets. Then training/education is needed.
How would eliminating welfare cause an already employed welfare receipient to seek a lower paying job?
Your logic is backwards. Let's work through this again, step by step.
1) most people who receive welfare already work. Most forms of welfare have actually required work ever since it was reformed during the Clinton administration. the earned income tax credit requires earned income (work).
2) Means tested welfare makes low wage workers more comfortable, thus they have little incentive to seek higher paying employment.
3) Low wage paying employers only pay low wages because they can find ample employees who are willing to accept a low wage. the reason that there are lots of people willing to accept a low wage is because those people are receiving means tested welfare (and if they accepted a higher paying job their welfare benefits would be reduced or eliminated)
4) If we eliminated welfare, we would not have more people in the job market, because most welfare receipients are already in the job market.
5) If we eliminated welfare, low wage employees would not be as financially comfortable, and thus they would be more likely to seek higher paying employment. Low wage employers would then have to compete harder for workers, and wages would tend to rise.