A reciprocal is simply dividing something into one. That something cannot be expressed as anything else. So Ho is a distance speed something, so to suit your agenda you choose to only preserve the time aspect. That's either CHEATING or ERROR or BOTH, is it not? So can miles per hour be expressed as hours in the reciprocal?? A reciprocal of anything must be convertible back to the original "something", whatever that was. I should not have to guess the orignal, should I?
Do you have a problem in using the quote function?
What is evident in your response you do not understand dimensional analysis or even simple arithmetic.
Here is a primer on dimensional analysis.
A physical property can be broken down into more fundamental units such as L (length), T (time) and M (mass).
Velocity which is distance divided by time is defined as LT⁻¹, acceleration LT⁻², force MLT⁻², pressure (force/area) MLT⁻²L⁻² = ML⁻¹T⁻² etc.
Since Hₒ is in km/s/Mpc it has the units LT⁻¹L⁻¹ = T⁻¹, hence 1/ Hₒ = T which is units of time.
Simple isn’t it and there is nothing nefarious about it.
So where do the units of Hₒ come from in the first place?
Astronomers are able to calculate recession velocities using the cosmological redshifts obtained from the spectra of distant galaxies and their distances when type 1a supernovae occur.
When plotting recession velocities in km/s against distance in Mpc they obtain this.
The line of best of fit is defined by the equation Velocity = Hₒ x Distance which is Hubble's law where Hₒ is the gradient or rate of change defined as km/s/Mpc.