As a landlord can I enter my home if I notify the tenant that I think a home repair is necessary and the tenant is unavailable?

You should not be entering the leased premises without the permission of the tenant unless there is an emergency. Once you provide advance notice to the tenant that you need access to the premises, s/he must not unreasonably deny you access.

However, I think because a water leak can do substantial damage to the premises if unaddressed and because the tenant specifically asked that you fix it, you can probably feel a little safer entering the premises. You should simply let the tenant know in writing that it represents a danger to the premises and that you intend to come fix it at exactly [time] on [date].

If you are renting a property 500 miles away, you might consider a rental management company who can address these issues locally.

I thought that when a tenant reports a repair that is required, that "invites" the LL in to make the repairs and/or see what repairs are needed.

Convenient time BS! A bathroom leak can seep through the floor and cause significant structural damage. Of course if your tenant refuses entry it's no the kind of thing you can start world war three over. But you cvan get an emergency order in landlord-tenant court allowing youto enter to do the repairs.