It's best not to let it go to seed.  This could actually result in a thinner lawn rather than a thicker one. Most lawn grasses, including Kentucky Bluegrass, spread through roots and stems; this is what thickens the lawn.  When grass grows tall enough to produce seed, its energy goes into the seed and not into the roots.  The end result is most likely a thinner lawn.  Also, the process can take up to four months.  For more information on this, see https://www.canr.msu.edu/news/when_grass_produces_seedheads

Raise your mower about an inch in summer.  This does protect the grass.   For more information, see https://extension.umn.edu/lawncare/mowing-practices-healthy-lawns