Hi, I hope my answers can give you a little more insight into your questions.
1. How can we plan, prioritize and manage such competing initiatives? 2. Is there a model to quantify technical debt and its impact to the business?
In order to plan and prioritize, you must have an understanding of the scope of the work that the technical tool/software/etc will cover. Secondly, prior to planning you would have performed the "Discovery analysis" of the different solutions you could implement that would be best suited in addressing your clients technical needs. During that discovery analysis, you following information you should have been able to obtain would the be pricing of the tool/software, what is the current/latest version of that tool/software, the amount of effor t you would need to set up and install, and then the timing of the next patching, upgrades, version release, etc. Upon that discovery analysis, you can then implement a technical plan that will outline how to and when to procure the next version release, patches, upgrades and so on. Then once you have create such plans, you can implement it into the Integrated Master Schedule (IMS) (if your project/program has one) or implement your own way of managing a timeline to ensure you procure the tools/softwares you need and present NOT only to the business (or PMO) department but to the client as well to ensure that all eyes can see it.
Once that has taken place, your management of ensuring to procure those needs mentioned about will come out of your Technical plan and your established timeline. Now obviously, this can get challenging when there comes to resource constraints and delays, but with a simple project schedule or even a timeline built into a excel file will assist you as a technical lead to communicate your needs.
To quantify technical debt (to me is a mixture of finding in the discovery phase and during execution phase), you can incorporate a mitigation strategy of how and when you identify technical debt and address that technical debt in your technical management plan. That way you are able have a "model" of how to address those issues. You can also incorporate the use of implementing risks that would show that the project/program timelines and deliverable obligations will be affected if technical debt is not resolved. It is also good to have strong communication chains with your project/program manager as well as with the client. The most important strategy of all when things get higher priority than your own is to Document, document. document. Whether that be through email, the project plan that you built for your team only or even in the Integrated Master Schedule for the whole project/program to ensure that you have covered your bases in the lines of communicating your need. Also the first is more preferable than the second, there are multiple ways to mitigate and address.
I hope I have been able to answer your questions. Please feel free to reach out to me if you would like to have any further discussions on the topic.