As you mentioned that you are not using the date picker feature so you are instructing the users to enter date of birth in input text in the format of YYYY-MM-DD and field type is Date. Right ???
The Date field type of EE will always require time also (HH:MM) so you can apply some JavaScript stuff which will amend the date value with time. For example: an user is entering date of birth as 1980-01-31 so you have to put some JavaScript so that it changes the date of birth value to 1980-01-31 12:00PM before submit. Rest of the things will be performed by EE.
Also, you can populate that date of birth in YYYY-MM-DD format by passing "format" parameter with the field.
You mentioned that you are using Low Search module and want to implement a search by age feature.
You can achieve it by adding Range Filter (http://gotolow.com/addons/low-search/docs/tags#form-tag-ranges). You can see within the doc that Low Search automatically detects the date field type of the channel and perform search as well.
For range filter, you will need to have this field something like:
<select name="range:date_of_birth_field">
<option value="">All</option>
<option value="<?php echo date('Y-m-d', strtotime('-20 years')).';'.date('Y-m-d', strtotime('-25 years')) ;?>">20 - 25</option>
<option value="<?php echo date('Y-m-d', strtotime('-25 years')).';'.date('Y-m-d', strtotime('-30 years')) ;?>">25 - 30</option>
<option value="<?php echo date('Y-m-d', strtotime('-30 years')).';'.date('Y-m-d', strtotime('-35 years')) ;?>">30 - 35</option>
<option value="<?php echo date('Y-m-d', strtotime('-35 years')).';'.date('Y-m-d', strtotime('-40 years')) ;?>">35 - 40</option>
<option value="<?php echo date('Y-m-d', strtotime('-40 years')).';'.date('Y-m-d', strtotime('-45 years')) ;?>">40 - 45</option>
<option value="<?php echo date('Y-m-d', strtotime('-45 years')).';'.date('Y-m-d', strtotime('-50 years')) ;?>">45 - 50</option>
<option value="<?php echo date('Y-m-d', strtotime('-50 years')).';'.date('Y-m-d', strtotime('-55 years')) ;?>">50 - 55</option>
<option value="<?php echo date('Y-m-d', strtotime('-55 years')).';'.date('Y-m-d', strtotime('-60 years')) ;?>">55 - 60</option>
<option value="<?php echo date('Y-m-d', strtotime('-60 years')).';'.date('Y-m-d', strtotime('-65 years')) ;?>">60 - 65</option>
<option value="<?php echo date('Y-m-d', strtotime('-65 years')).';'.date('Y-m-d', strtotime('-70 years')) ;?>">65 - 70</option>
</select>
You can see, I am using PHP code to output the proper range for the date of birth so you can to enable PHP for that template.