I think once we have a national HSR network, we should break up each LD route at each station along that LD route which is also a HSR stop, and then reschedule the broken up trains so that they start in the early morning and finish their runs by the late evening.
One of the benefits of LD trains is that you can catch one in the evening, sleep for the night, and wake up at your destination. If you do a morning to evening run you've spent a day of conscious hours on travel... not so good.
If you have a conventional speed daily train with a 12-18 hour run from one HSR station to another HSR station with a bunch of intermediate stops that are served only by this daily train, you either get to make it a day train only, or force some people to board in the middle of the night. Do you really think boarding at 3AM is an improvement over daytime travel?
Also, consider how you get from Connersville, IN to McCook, NE. If we preserve the current long distance schedules, you can board at Connersville around 3 AM, reach Indianapolis around 4:45 AM, perhaps board a HSR train around 6:00 AM, and if its top speed of 220 MPH works out to an average of 150 MPH after you account for the travel through cities to downtown stations at slower speeds, the almost 1100 miles would take a little over 7 hours, so you'd arrive in Denver a little after 1 PM. Then you could wait around until 8:10 PM to board the eastbound Zephyr, and get to McCook a little before 1 AM.
Wouldn't it be better to board a day train at Connersville at 4:45 PM, reach Indianapolis at 6 PM, board a HSR sleeper train that provided early boarding and departed around 10:00 PM and reached Denver around 5:30 AM, and then lingered at the platform for several hours so that you could disembark at your convenience to transfer to a train that would depart Denver at 9:00 AM and reach McCook at 1:40 PM?
(Or maybe you change trains at Omaha instead of Denver and take the westbound Zephyr instead, but I think in that case the travel time works out roughly the same.)
Another thing to consider is that since Connersville is the only stop between Cincinnati and Indianapolis currently, if there were a train that ran from Cincinnati to Connersville to Indianapolis and that was its entire run, adding some more intermediate stops would probably be viable in a way that isn't when the train runs all the way from Chicago to New York City.
If we had enough sleeping cars, the other alternative is an overnight train with a set out sleeper for each intermediate stop. You end up needing two cars per stop to make that work, one carrying passengers towards the HSR station each night, and another carrying passengers away from the HSR station each night (unless there's enough demand at each stop to need more than that).