It's hard to answer your questions without really knowing the study system in great detail.
The ASC model helps to alleviate a very specific kind of ascertainment bias - that which derives from including only
variable sites in your analysis.
To me, the more important issue with the analysis you've described is that you're assuming that a single tree exists for all the 10K accessions you're studying. From what you've written, it sounds like you have ~5K SNPs for each of ~10K terminals from ~10 species total. I see no good reason to assume that the 5K SNPs should follow a single tree within each species, nor (assuming that some introgression and/or ILS is likely between the species in your sample) between species.
It seems to me that it would be more appropriate in this case to use methods like SNAPP, SVD-Quartets or SNP-based network approaches like SNAQ.
Of course, the best method really depends on the questions you're asking and a deep understanding of the system and the data. You're best placed to make those calls!