I'm absolutely positive that it will be impossible to get to Santa Cruz on trail. There are no east-west trails in California that will get you there.

Whether the Pacific Crest Trail is accessible depends on time of year. Unless you are a dedicated mountaineer with snow gear it would be folly to attempt a trans-sierra passage on foot between November - April, unless one is walking over Tehachapi Pass. Majority of the trails over mountains will be under snow for a large part of the year. As will all the high passes with roads. Crossing the Rockies anywhere but roads would be a real adventure for sure, unless timed correctly.

Hiking on the trails would be a wilderness permitting headache, too. Too many jurisdictions, got to sort them all out, some places you have to apply for permits months in advance and need specific dates, which are probably not possible for such a trip. Not to mention proper food storage in the Sierra Nevada. Other parks and wilderness areas across the country have varying food storage regulations as well.

Unless you're going to do this journey in shorter legs, timed to work with the seasons/weather patterns, it's just not gonna be possible, IMO. Timing it so the crossing of the mountains doesn't happen in midwinter but the desert crossings don't happen in summer sounds like a few airplane tickets back and forth. Specially if you can't maintain a good pace.

2 mph for 5,000 miles? Yikes.