Okay here are the results of my tests. Each grouping of active themes and plugins was tested using a 3-run test on webpagetest.org, and the result links are included below.
Types, Views, 2017 theme
Median TTFB: 2.8s
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Types, Views, Mychurchwebsite theme
Median TTFB: 1.327s
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All Toolset, Mychurchwebsite theme
Median TTFB: 1.752s
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All Toolset, Mychurchwebsite theme, Other plugins A - E
Median TTFB: 3.628s
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All Toolset, Mychurchwebsite theme, Other plugins A - M
Median TTFB: 4.201s
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All Toolset, Mychurchwebsite theme, Other plugins A - S
Median TTFB: 4.662s
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All Toolset, Mychurchwebsite theme, All Other plugins (all those that were originally active)
Median TTFB: 5.370s
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My comments:
1. Types and Views with your theme is actually quite fast - faster than Types and Views with 2017. As other plugins are activated, you can see the TTFB performance takes incremental hits. This is to be expected to some degree, because more plugins means more code execution. The biggest jump I see is in activating plugins A - E in alphabetical order, which produced a jump around 2 seconds in loading time. You could try to run some more pointed tests to see if there is one plugin causing the majority of that difference, and consider switching to another plugin. Or if it's critical, we may be able to collaborate with their developers on a compatibility fix.
2. The tests didn't come close to a 20 second TTFB reproduced in any scenario. However, I was able to reproduce longer TTFB times by manually loading the site in the browser with multiple concurrent tabs open and loading. However, you can see from the tests that there are times when the site performs much much faster. The code and content of your site isn't changing enough between tests to justify that kind of time difference between our results. So I'm not sure how to explain those differences. Check with your hosting company to see if they have object caching, varnish, memcache, redis, or any server-side caching system in place. If so, see if they will disable it temporarily and test again. It's counterintuitive, but if there is a conflict between Toolset and their caching system it will actually speed up performance.
3. Consider adding a third-party WordPress caching plugin. WP caching plugins, unlike the server-side caching systems I mentioned before, typically work by creating static HTML files and serving those up instead of making expensive queries for each request. This can be really helpful on pages where you have lots of nested Views and templates, and sites with lots of plugins. Custom search results performance isn't usually improved, though.
4. If you want to pinpoint slow DB queries, consider a plugin like Query Monitor. If any single Toolset DB query is taking longer than 0.5ms consistently, we can investigate in more detail.
Please let me know your thoughts and we can go from there.