Hybrid setups generally fall into two main scenarios:
- Hybrid is in place while working on completely migrating to SharePoint Online. This eventually results in the deprecation of the on-prem farm.
- There are requirements that prevent you from entirely from migrating to SharePoint Online, but you want to be able to take advantage of the services provided by the cloud.
You are describing the second case and point out a few of the limitations of running on an external service. The SharePoint Online software boundaries and limits article provides an overview of some of the basic structures that are limited in the cloud environment. For example, the max file upload is 10GB and list views are limited to 5000 items. Now, over time, as the service improves, those limitations will be increased. The max file size able to be upload used to be 2GB and site collections used to be limited to 1TB instead of the 25TB today.
For the issues that you list, the first and second are about new paradigms and the second and third are about control. Let's address each on individually.
SharePoint development has gone through several iterations. First, you had Farm Solutions (a.k.a. full trust code). These grant the code full access to the farm and is not throttled or managed by the system in any way. In a shared service like SPO, having random code running around can be detrimental to the service. On-prem, you know exactly what is deployed to your farm and you know how to fix it. SPO tried to get around this by introducing Sandbox Solutions, which gave code limited access to the farm, but was eventually deprecated in favor of other models. These other models include use of CSOM or SharePoint add-ins.
A great resource for building apps or add-ins using the latest methods is the Office Patterns and Practices site as well as their GitHub site with samples. Depending on your requirements for applications, you can decide if you need to greater control of Full Trust on an on-prem farm or if you can use the solutions provided by the PnP to do what you need. The SharePoint Store also provides a collection of add-ins by third parties that may meet some of all of your needs.
SharePoint Admin Interactions
The PowerShell changes are partially an example of control changes (see more below), but part of it is also about what even needs to be managed. SharePoint on-prem requires that you set up physical servers, make sure they are healthy, manually create and manage web apps, etc. SPO allows for you to avoid having to think about the bare metal and allows for higher level conversions. Instead of having to work with IT and finance to plan server purchases, you can focus on information architecture and how and who has access to what.
Trade-offs of Control
Loss of access to the error logs and some of the more advanced PowerShell functionality is in some cases better. Running an on-prem server means that you have to do updates, make sure the update works, make sure there are enough machines available to service requests, and much more. You need the error logs and heightened control to resolve issues. The service is provided in such a way that it removes your need to debug these sorts of things. Error logs aren't needed when you personally won't need to fix the problems. It is similar to the reason why users of your on-prem deployment don't need to access the logs. There are Service Level Agreements that guarantee certain uptime and functionality of the service so you should ideally not be hitting errors or having site downtime.
What this means for you is that you can spend more time working on your core business and less time making sure your tooling works.
Bringing it home; What do I do?
Choosing to run a hybrid solution depends on what both the comfort level of having someone else run your infrastructure (which by having an SPO tenant you are already at least somewhat comfortable with) and what sort of integration points are needed. Hybrid allows for certain features (OneDrive for Business, Hybrid App Launches for SPO only features like O365 Video, etc.) to be offloaded to SPO while certain other functionality (e.g. team sites on-prem that have intense full-trust solutions and customizations) can be retained on your on-prem farm. If you do decide to go for hybrid, there are several guides available for you as well as free eBooks for planning and configuring hybrid.
If you want to try it out, set up a trial 2016 farm with hybrid and see if matches your needs and expectations. Hands on experience can flesh out details of your decision and can help expose other issues or concerns you may have with running an on-prem farm
SPO, on-prem, and hybrid are all great tools for document lifecycle management and collaboration. You can find some nice high level slides in the SharePoint Hybrid Courseware and Curriculum. While it's a bit old (from 2014), the basic considerations that you brought up are addressed and gives you some decks to help explain hybrid to other people in your organization and your choice going forward.