This is why Jesus said "you cannot work for God and money at the same time without cheating on one or the other".
Brother, your exegesis here is just not consistent with the scriptures. We have to take in the whole counsel of God before we make a doctrine out of one verse, and there are a few scriptures which contradict your interpretation of Matthew 6:24. For instance Paul tells us in 2 Thessalonians 3 that he forsook the wages he was entitled to for preaching the gospel, as he did at many other churches, and instead he worked for the money to pay for his own bread:
2 Thessalonians 3:8 nor did we eat anyone's bread without paying for it, but with toil and labor we worked night and day, that we might not be a burden to any of you.
Was Paul following a "convenient doctrine" here? No, he was laboring for wages so he wouldn't be a burden on anyone. This is why Paul gave this command:
2Th 3:10 For even when we were with you, we would give you this command: If anyone is not willing to work, let him not eat.
2Th 3:11 For we hear that some among you walk in idleness, not busy at work, but busybodies.
2Th 3:12 Now such persons we command and encourage in the Lord Jesus Christ to do their work quietly and to earn their own living.
If you won't work, you don't eat. Pretty straightforward isn't it? We as Christians are told here to earn our own living and not be a burden to anyone. This is quite a contradiction to your doctrine that you can't serve God and earn money at a job at the same time. Scripture is telling us that if we are a burden to others, depending on others for sustenance when we could be earning wages to take care of ourselves, we are outside of Gods will because we are walking in idleness. Such people should not be fed, by the command of the apostle Paul.
We also see Paul telling us in 1 Timothy 5:8 that if we don't provide for our families we are worse than unbelievers:
But if anyone does not provide for his relatives, and especially for members of his household, he has denied the faith and is worse than an unbeliever.
So unless scripture is contradicting itself, we can see that your interpretation is not accurate.
What Jesus is talking about in Matthew is serving a false idol versus the true God. The dichotomy is that you are putting money in the place of God, and serving money instead of God. According to the word of God, merely working to earn a living is not putting money in place of God. It's actually doing what God commanded us to do through the apostle Paul.
A person working solely to earn more and more money, to use to increase his material wealth and status, is serving an idol. A person working to put food on the table for his family is serving God.
You say that a person who is working couldn't necessarily spend all of their time witnessing for God therefore they are outside of Gods will. Well, if that's true then you should only sleep 3 hours a night so you can get out there on the street and tell more people about Christ. You are wasting all of that time sleeping when you could probably go at least 6 months on that schedule.
Do you see then that there is a time and a place for everything? How do you know that God doesn't want a person at a certain job so they can reach a certain person, even if they only get to talk about Christ 5 minutes a week? That's the point, you don't know that, so you can't condemn someone for supposedly wasting their time working when they are doing Gods will by representing Christ at their work place the way God wants them to.