This is the kind of question where proper answer can take up a small book or at least a whitepaper. But yes, your challenges are all quite common and it's issues like this that motivated me (and many others) to build their own framework on top of QuantLib.
So while I cannot give an exhaustive answer, here are some tidbits which I hope help:
What are the best practices for maintaining and updating interest rate curve bootstrapping models, especially in a small team setting?
You want to reduce your configuration to a few settings that end users can quick change. This will typically require to build instrument symbology and instrument database so end user can quickly decide on a list of instruments to use for curve building by picking them out of a set - like being able to pick things like ICE.SR3.M25 out of a dropdown if one wants to use interest rate future for the curve build. Similarly, you want to expose as configuration things like interpolation methods so they can be changed easily. It might end up looking something like this.
Is it common in the industry to manually adjust these models in response to changing market conditions, or is there a trend towards more automated processes?
It is a bit of both. Those shops that actively trade off the curve have to pay great attention to curve quality, and they will have a quant trader eyeball the curve regularly throughout the day and adjust data and settings as needed to make it look good. If you are a smaller shop that trades securities that are less sensitive to curve quality, or a risk management team, it is better to just set up the curve in a way that is very robust and unlikely to fail to calibrate, and then leave it to the automation. Example of that would be to just use IR swaps spaced by some months / years and a flat rate interpolator, this will be quite stable day to day.
For those using QuantLib or similar libraries, what strategies or tools are recommended to improve the robustness and reliability of bootstrapping methods?
As per above, it actually comes down not so much to the specific library you are using but to the choice of instruments and calibration methodology. If you are using spline interpolation with lots and lots of instruments you will have a very flaky calibration that is also sensitive to market inconsistencies or liquidity issues, and such curves will require a lot of babysitting. One the other hand, if you have a simplistic setup with sparse instruments and flat rate interpolation, your pricing of off-market instruments will not be that good. So you will want to find a trade off between those two extremes that is right for your organization.
