The idea of praying to God regularly is to build your trust and faith in him. How can you do that if you are reciting someone else's prayer? Philippians 4:6
"Do not be anxious over anything, but in everything by prayer and supplication along with thanksgiving, let your petitions be made known to God"
"Draw close to God, and he will draw close to you. Cleanse your hands, you sinners, and purify your hearts, you indecisive ones."
Prayer to God should always be from your heart. You cannot cultivate a relationship with God by merely reciting words that really have no meaning to you. Doing so takes away from the real intention of prayer. The Jew's made the mistake of doing things out of habit and routine. That is not what God want's from us and even warns against it at Matthew 6:7-8
"When praying, do not say the same things over and over again as the people of the nations do, for they imagine they will get a hearing for their use of many words. So do not be like them, for your Father knows what you need even before you ask him."
You can look at those prayers in order to gain an understanding of what is proper to ask for. Even the prayer that Jesus gave us in Matthew 6 is just a model.
I think the purpose of using the psalms of the day is to get people to read the bible and to keep them thinking about spiritual things. Doing so is a good precursor to making regular prayer and maybe that is the goal of this schedule. Good motive, but what the bible teaches us to do is make our own regular prayers to God. 1 Thessalonians 5:17-18
Pray constantly. Give thanks for everything. This is God’s will for you in Christ Jesus.