Do bigger buttons improve usability and how they correspond with design? [case]

We're trying to incorporate a style guide which helps developers understand and build CSS across different resolutions. After reading these:

  1. https://uxplanet.org/buttons-in-ui-design-the-evolution-of-style-and-best-practices-56536dc5386e (Nick Babich)
  2. https://builttoadapt.io/intro-to-the-8-point-grid-system-d2573cde8632 (8pt grid system)
  3. https://uxplanet.org/7-rules-for-mobile-ui-button-design-e9cf2ea54556
  4. https://developer.apple.com/design/tips/
  5. Bootstrap, Foundation, iOS and Material Design guidelines

I came to a question do bigger buttons on 1920x1080 (desktop) and above improve usability and do they ruin or fit into design at the same time? Is there any standard rule/research on this topic or does it apply only to particular scenarios?

[CASE] Our case is a website for a car dealer whose main clients group is consisted of 40+ years old businessmen leasing the vehicles. On the other hand, we want to build a site that also attracts younger group so poor eyesight shouldn't be a factor. In the end, we get to the question above - does it improve the design in terms of usability or not?

(the red frame is a full HD viewport)

buttons dimension

buttons compared