What the priest is reading
A muhurtham is not picked from a printed list. The priest reads the panchangam for the day's tithi, nakshatram and lagnam, and weighs those against the nakshatram of the head of the household. The result is usually a window of an hour or two on a particular morning rather than a whole day.
Why most ceremonies are in the morning
Because the window generally needs to fall before Rahu Kalam. That is why Griha Pravesham invitations so often give a narrow time rather than a start time, and why guests are asked to arrive earlier than they expect.
Family practice narrows it further
Beyond the panchangam, families follow their own conventions about which Tamil months suit functions of this kind, and these differ from household to household. This is one area where asking an elder in your own family is more reliable than any general calendar online.
The practical problem nobody warns you about
Auspicious dates cluster. In a given month there may be only a handful, and within those only a couple of workable morning windows. Every family in the city is reading the same panchangam, so those mornings are booked early and experienced priests go first.
Questions worth asking your priest
Which windows are available in the months you are considering. Whether your situation is a first entry, a return after absence, or a re-entry after construction, since that changes what is performed. How long the sequence you have chosen will actually run, so you can tell guests a realistic time. And whether anything in your family's practice rules out particular months.
If your date depends on a builder
Treat the muhurtham as the fixed point and the handover as the variable, not the other way round. Families who wait for the handover to confirm before booking the priest routinely end up with a worse date.
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