Argo improves one specific aspect for page load speed = server response time. See my Argo experience at https://community.centminmod.com/threads/cloudflare-argo-smart-routing-in-action.17517/ tracking real world improvements via Google Analytics and Google Datastudio.
Thus in theory it indirectly improves Time To First Byte (TTFB) metrics which indirectly can improve LCP. Note though a fast TTFB doesn’t necessarily mean a fast LCP but a slow TTFB will make LCP slower. This is because there are other factors that can impact LCP other than TTFB. This is outlined in the link you posted at Optimize Largest Contentful Paint | Articles | web.dev
The most common causes of a poor LCP are:
Render blocking javascript/css due to 3rd party advertising can make results vary greatly between test runs too as the weight of javascript can vary depending on ads served.
You can use various page speed tools to test. I did a writeup up at https://community.centminmod.com/threads/google-page-speed-insights-and-google-core-web-vital-metrics.20735/ you may want to read.
However, if you’re measuring Core Web Vital metrics in real world i.e. Google Chrome User Experience (CRuX) reports via Google Search Console, then your results can be impacted by client side rendering too. I posted an example at Performance Tutorials - Google PageSpeed & Webpagetest.org - #4 by eva2000
For my forums, majority of the threads are picking up the title bar’s H1 tag as the LCP item. So thought I’d use Element Timings API to measure when my title bar’s H1 tag actually renders for me in the real world. FYI, Google Search Console Vital metrics reported my forum index having a LCP of 1.5s for 75% of visits which would of been made up of both guests + logged in members.
For forum index page measured H1 title bar rendered times for me in Brisbane, Australia hitting my forums in US West Coast behind Cloudflare
- Logged in member/visitor no CF full HTML page cache between 2000-2500ms so ~2-2.5 seconds
- Logged in member/visitor no CF full HTML page cache but watching a video on my computer blew out the H1 tag render time to 19790ms ! Yes 19.79 seconds !
- Guest logged out visitor with CF full HTML page cache between 900-1300ms with random variation spikes of 2400-4900ms (could be factors local to my PC/apps in use)
This goes to show in real world, your visitors maybe also multi-tasking on their computers which may impact their browsing page load experience and thus impact your real world Google Search Console Speed LCP/FID metrics and Google Page Speed Insight’s reported field/origin metrics
Now as to CF Load balancer, this will only help with regards to LCP if it improves server response time (TTFB) from every country your visitors come from. Meaning you’d need to have as many origin servers serving your content in all regions that CF operates in to achieve an improvement in server response time for CF to origin connection.
This is because Cloudflare CDN cache doesn’t cache dynamically generated HTML by default unless you tell it to do so https://support.cloudflare.com/hc/en-us/articles/200172516-Which-file-extensions-does-CloudFlare-cache-for-static-content- - which you do via cache everything rules.
Optimal bang for your buck for page speed would be Cloudflare Business plan’s bypass cache on cookie + cache everything page rules or via CF worker bypass cache on cookie for guest based full HTML page caching https://blog.cloudflare.com/caching-anonymous-page-views/ and combine that with Argo 
However, this only improves CF CDN cached responses. For non-cached cache MISS requests they still hit your origin server. So you’d need to ensure your origin server is optimally configured.
To fully optimise you need to optimize 3 segments.
- segment 1 - connection between visitor and CF edge server i.e. CDN cache, WAF, Firewall, Page Rules, Mirage, Polish webP, HTTP/2, HTTP/3, CF Workers (i.e. custom/advanced caching) etc
- segment 2 - connection between CF edge server and your origin i.e. Argo, Railgun & Full SSL/ECDSA SSL certificates and optionally Load Balancer provided you have origins in every country that your visitors come from.
- segment 3 - your origin server’s performance/optimisations i.e. web server, PHP, MySQL server optimisations and server hardware specs.
Cloudflare can only help for segments 1 & 2 for cached guest/non-logged in based visitors.