First of all, it should be clear that the Glorious Qur’an and the purified Tradition (Sunnah) of the Prophet, peace and blessings be upon him, are the references of every Muslim for the realization of the rules of Islam. The Qur’an can be understood according to the principles of the Arabic language without affectation or controversy, and the Sunnah can be acquired by reference to the trustworthy transmitters of Hadith (collected sayings of the Prophet). Therefore, the Qur’anic verse your referred to in your question should be understood according to the rules of the Arabic language as illustrated by the prominent Muslim scholars.

Dr. Muzammil H. Siddiqi, president of the Fiqh Council of North America, states the following: The word “salah” literally means “to come close and near”. It is used in the Qur’an for prayers as well as for blessings. Through our salah or prayers we try to come closer to Allah with submission and surrender. Allah also draws near to His Prophets and the Believers through His blessings. In the Qur’an it is also mentioned that Allah sends “salah” on the Prophet (Al-Ahzab 33: 56) as well as on the believers (Al-Ahzab 33: 43). When Allah comes closer to His Prophets (peace and blessings be upon them all), and to other believers, it means that He comes closer to them with His blessings, love and favor. Allah says in the Qur’an, [Lo! Allah and His angels shower blessings on the Prophet. O ye who believe! Ask blessings on him and salute him with a worthy salutation.] (Al-Ahzab 33: 56) and [He it is who blesseth you, and His angels (bless you), that He may bring you forth from darkness unto light; and He is Merciful to the believers.] (Al-Ahzab 33: 43)

Thus, when it is said that Allah sends His salah on the Prophet (peace and blessings be upon him) it means that He blesses the Prophet. When it is said that angels send salah, it means that they pray to Allah for His blessings for His Prophet. When we, the believers, are told to do so it means the same thing that we should ask Allah to bestow His blessings on Prophet Muhammad.
Allah sent and continues to send His blessings on the Prophet (peace and blessings be upon him) and the believers. His blessings are countless and they come in numerous ways. We should pray to Allah to send more and more blessings on Prophet Muhammad. The Prophet Muhammad himself has instructed us that when we send blessings unto him, we should say “Allahumma salle `ala Muhammad wa `ala ali Muhammad kama sallaita `ala Ibrahim wa `ala `ali Ibrahim. Innaka hameedun Majeed.” This and a number of other similar versions of salath`ala al-nabee are mentioned in many authentic Hadiths. In our prayers we should use only those authentic versions and should not make any changes from our side.
Allah told us that we should send salahon Prophet Muhammad (peace and blessings be upon him). The Prophet (peace and blessings be upon him) then told us that the best way to do that is to request Allah to send His blessings upon him. So, there is no shirk. We are not sending blessings on him. Prophet Muhammad (peace and blessings be upon him), wanted to include his Aal also in this blessing. The word Aal has various meanings. It could mean his family as well as his followers.