It would only need to be "these people" if Paul was listing specific people. Let's look at what he says.
Titus 2:1-10 But as for you, speak the things which are fitting for sound doctrine. Older men are to be temperate, dignified, sensible, sound in faith, in love, in perseverance. Older women likewise are to be reverent in their behavior, not malicious gossips nor enslaved to much wine, teaching what is good, so that they may encourage the young women to love their husbands, to love their children, to be sensible, pure, workers at home, kind, being subject to their own husbands, so that the word of God will not be dishonored. Likewise urge the young men to be sensible; in all things show yourself to be an example of good deeds, with purity in doctrine, dignified, sound in speech which is beyond reproach, so that the opponent will be put to shame, having nothing bad to say about us. Urge bondslaves to be subject to their own masters in everything, to be well-pleasing, not argumentative, not pilfering, but showing all good faith so that they will adorn the doctrine of God our Savior in every respect.
So what types of people do we have? Older women. Younger women. Older men. Younger men. Bondslaves.
Then he says:
Titus 2:11-14 For the grace of God has appeared, bringing salvation to all men, instructing us to deny ungodliness and worldly desires and to live sensibly, righteously and godly in the present age, looking for the blessed hope and the appearing of the glory of our great God and Savior, Christ Jesus, who gave Himself for us to redeem us from every lawless deed, and to purify for Himself a people for His own possession, zealous for good deeds.
So he lists these TYPES of people and the instruction to give them. Then he says "For (because) the grace of God has appeared to all people (not just a select type, but all types)". And he lists what this grace does; brings salvation (not just potential) and instruction.
If your view was correct, then every single person would be purified from every lawless deed, and be a people for His own possession, zealous for good deeds. God's grace actually DOES something.