Is it considered a sin not to take the initiative to go to the mosque for prayer with guests or to lead them in prayer at home due to embarrassment, and what should be done about it?

You should have invited your guests to pray with you in the mosque to earn reward and to encourage goodness, and your shyness towards them is not commendable. If going to the mosque was not possible, you should have sought the virtue of congregational prayer by praying at home. Delaying prayer beyond its prescribed time is not permissible except for a legitimate Shar'i excuse, and the presence of guests is not an excuse. If you delayed the prayer until its time expired for this reason, you must repent to Allah and make up the prayer if you have not already performed it.